Exploring Electron and Phonon Transport at the
Nanoscale for Thermoelectric Energy Conversion
by
Austin Jerome Minnich
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
ARCH~/ES
MASSACHUSET INSTJ
OFTECHNOLOGY
Doctor of Philosophy
JUL 2 9 2011
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
June 2011
@ Massachusetts Institute of Technology 2011. All rights reserved.
Author ...............
Department of Mechanical Engineering
May 20, 2011
Certified
by..........
.
.......................
......
Gang Chen
Carl Richard Soderberg Professor of Power Engineering
Thesis Supervisor
A ccepted by ...........................
David E. Hardt
Chairman, Department Committee on Graduate Students
'2
Exploring Electron and Phonon Transport at the Nanoscale
for Thermoelectric Energy Conversion
by
Austin Jerome Minnich
Submitted to the Department of Mechanical Engineering
on May 20, 2011, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Thermoelectric materials are capable of solid-state direct heat to electricity energy
conversion and are ideal for waste heat recovery applications due to their simplicity,
reliability, and lack of environmentally harmful working fluids. Recently, nanostructured thermoelectrics have demonstrated remarkably enhanced energy conversion efficiencies, primarily due to a reduction in lattice thermal conductivity. Despite these
advances, much remains unknown about heat transport in these materials, and further efficiency improvements will require a detailed understanding of how the heat
carriers, electrons and phonons, are affected by nanostructures.
To elucidate these processes, in this thesis we investigate nanoscale transport using
both modeling and experiment. The first portion of the thesis studies how electrons
and phonons are affected by grain boundaries in nanocomposite thermoelectric materials, where the grain sizes are smaller than mean free paths (MFPs). We use the
Boltzmann transport equation (BTE) and a new grain boundary scattering model to
understand how thermoelectric properties are affected in nanocomposites, as well as
to identify strategies which could lead to more efficient materials.
The second portion of the thesis focuses on determining how to more directly measure heat carrier properties like frequency-dependent MFPs. Knowledge of phonon
MFPs is crucial to understanding and engineering nanoscale transport, yet MFPs are
largely unknown even for bulk materials and few experimental techniques exist to
measure them. We show that performing macroscopic measurements cannot reveal
the MFPs; instead, we must study transport at the scales of the MFPs, in the quasiballistic transport regime. To investigate transport at these small length scales, we
first numerically solve the frequency-dependent phonon BTE, which is valid even in
the absence of local thermal equilibrium, unlike heat diffusion theory. Next, we introduce a novel thermal conductivity spectroscopy technique which can measure MFP
distributions over a wide range of length scales and materials using observations of
quasi-ballistic heat transfer in a pump-probe experiment. By observing the changes
in thermal resistance as a heated area size is systematically varied, the thermal conductivity contributions from different MFP phonons can be determined. We present
the first experimental measurements of the MFP distribution in silicon at cryogenic
temperatures.
Finally, we develop a modification of this technique which permits us to study
transport at scales much smaller than the diffraction limit of approximately one micron. It is important to access these length scales as many technologically relevant
materials like thermoelectrics have MFPs in the deep submicron regime. To beat
the diffraction limit, we use electron-beam lithography to pattern metallic nanodot
arrays with diameters in the hundreds of nanometers range. Because the effective
length scale for heat transfer is the dot diameter rather than the optical beam diameter, we are able to study nanoscale heat transfer while still achieving ultrafast
time resolution. We demonstrate the modified technique by measuring the MFP distribution in sapphire. Considering the crucial importance of the knowledge of MFPs
to understanding and engineering nanoscale transport, we expect these newly developed techniques to be useful for a variety of energy applications, particularly for
thermoelectrics, as well as for gaining a fundamental understanding of nanoscale heat
transport.
Thesis Supervisor: Gang Chen
Title: Carl Richard Soderberg Professor of Power Engineering
Acknowledgments
Many theses' acknowledgements begin with a statement like, "This thesis could not
have been completed without..." Having just finished this thesis, I can confirm that
this statement, while clich6, is absolutely true. Without the help of many people I
would not have been able to complete my thesis, and I'd like to gratefully acknowledge
their help here.
First, I would like to thank my advisor, Professor Gang Chen. I am very fortunate
to have been able to work for Gang. He routinely goes above and beyond what is
expected of an advisor, from meeting with students as often as they like, to reading
drafts of papers quickly and giving insightful suggestions, to teaching students how
to give a good presentation. I am particularly thankful to Gang for helping me to
develop as an independent researcher by giving me almost complete freedom to choose
my research topics. Without his help, I would not have been able to secure a faculty
position straight from graduate school.
Next, I'd like to thank my thesis committee members, Millie Dresselhaus, Keith
Nelson, and Evelyn Wang. All took significant amounts of time to give me advice
on my research and career paths, as well as to write reference letters for my faculty
applications. I'm particularly grateful to Millie for her quick response time and helpful
comments on the several papers we have written together over the years. Keith, and
members of his lab, have been great collaborators, and they have provided very useful
insight into my work with ballistic transport and ultrafast experiments. I am also
grateful to Evelyn for sharing her perspective as a young faculty member with me.
I'd like to thank several former members of the NanoEngineering group: Qing
Hao, Sheng Shen, and Aaron Schmidt. Qing Hao and Sheng Shen were extremely
helpful and kind to me when I first arrived in the group in 2006. Qing taught me
much of what I know about the practical details of running experiments. Sheng was
my officemate for three years and remains a great friend. Aaron and I started working
together two years after I arrived in the group, when I was learning to use the pumpprobe system. From the beginning, Aaron was very patient and spent quite a lot
of time teaching me how to use the system. After he returned to Boston following
his postdoc, he again spent a significant amount of time helping us fix the system so
that we could take reliable data. In particular, the silicon measurements of chapter
5 could not have been acquired without his help. Qing, Sheng, and Aaron will make
excellent professors and I very much look forward to working with them again as a
colleague.
I am very lucky to work with a great group of people, the NanoEngineering group
of MIT. The collegial atmosphere in our group made life at MIT much more enjoyable. My labmates were also a great resource, and I greatly benefitted from the
communication and collaboration with them.
I am happy to have worked with two members of Keith Nelson's group, Jeremy
Johnson and Alexei Maznev. I have really enjoyed discussing our joint work and know
that the quality of my work is higher because of my interaction with them.
I'd like to thank Kurt Broderick, a staff member at MTL; and Mark Mondol, who
runs the electron-beam lithography machine at MIT. Both provided invaluable help
in fabricating samples.
I'm grateful to the Department of Defense and the National Science Foundation
for their financial support of my PhD.
I am really lucky to have had a great group of friends during my time at MIT.
Their presence made my time here much more meaningful and enjoyable. In particular
I'd like to acknowledge my very close friends Josh Taylor, Allison Beese, Priam Pillai,
Eerik Hantsoo, Matthew Branham, Kit Werley, Janice Cho, Albert Leung, and Jack
Chu. I will really miss them as we all head out of Boston.
Finally, I am very grateful for my wonderful fianc6e, Song Yi Kim, and my family
members. There were many times during my PhD when I felt so discouraged and
frustrated that I wasn't sure I was capable of finishing. During these times, the encouragement and support of my family was really what kept me going and ultimately
enabled me to succeed. I am particularly indebted to Song Yi, and my mom, dad,
and sister, for their love and support.
Contents
21
1 Introduction
. . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.1
Nanoscale heat transfer and thermoelectrics
1.2
Outline of the thesis
2 Carrier transport in thermoelectric nanocomposites
27
2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.2.1
Relaxation time approximation
. . . . . . . . . . . . . . . . .
31
2.2.2
Additional details . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.2.3
Electron relaxation times . . . . . . . . . . . . . . . . . . . . .
35
2.2.4
Phonon modeling . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2.5
Summary of the calculation
. . . . . . . . . . . . . . . . . . .
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
. . . . . . . . . . . . . . . . . . . . . . . .
44
2.4.1
Phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.4.2
Electron scattering . . . . . . . . . . . . . . . . . . . . . . . .
46
Nanocomposite results . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.5.1
N-type nanocomposite Sio. 8 Geo. 2 . . . . . . . . . . . . . . . . .
54
2.5.2
P-type nanocomposite Sio.8 Geo. 2 . . . . . . . . . . . . . . . . .
58
2.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.3
Model validation
2.4
Modeling nanocomposites
2.5
7
3 Determining phonon mean free paths: the quasi-ballistic heat transfer regime
3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
The quasi-ballistic heat transfer regime . . . . . . . . . . . . . . . . .
3.3
Numerical solution of the frequency-dependent phonon Boltzmann transport equation . . . . . . . . . . . . . . . . . . . .
3.3.1
Calculation of the equilibrium distribution
3.3.2
Role of optical phonons . . . . . . . . . . .
3.3.3
Interface condition . . . . . . . . . . . . .
3.3.4
Phonon dispersion and relaxation times . .
3.3.5
Numerical details and boundary conditions
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Discussion . . . . . . . . . . . . . . . . . . . . . .
3.6
Unresolved puzzles . . . . . . . . . . . . . . . . .
3.7
Conclusion . . . . . . . . . . . . . . . . . . . . . .
4 Modifications to the pump-probe experiment
5
93
4.1
Background . . . . . . . . . . . . . . . . . . . . . . . .
93
4.2
Modifications to signal detection circuitry
. . . . . . .
95
4.3
Correction of laser beam astigmatism . . . . . . . . . .
98
4.4
Sample preparation and film quality . . . . . . . . . . .
101
4.5
Other miscellaneous modifications.........
102
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
. ..
102
Measuring mean free paths using thermal conductivity spectroscopy 105
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
105
5.2
Experim ent . . . . . . . . . . . . . . . . . . . . . . . .
108
5.3
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . .
112
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
115
8
6 Measuring mean free paths at the nanoscale
117
117
................................
6.1
Introduction ......
6.2
Heat Transfer Model ...........................
119
6.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2
Single Dot Heat Transfer Model . . . . . . . . . . . . . . . . .
120
6.2.3
Dot Array Heat Transfer Model . . . . . . . . . . . . . . . . .
121
. . . . . . . . . . . . . . . . . . . . .
123
6.3
Sample Design and Fabrication
6.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Summary and future directions
131
7.1
Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A Numerical constants used in the nanocomposite model
137
B Derivation of the relation between transmissivity and interface conductance
141
10
List of Figures
1-1
Figure of merit ZT of current state of the art thermoelectric materials
versus temperature. The dashed lines show the maximum ZT values
for bulk state of the art materials, and the solid lines show recently
reported ZT values, many of which were obtained in bulk nanostructured materials. (From Ref. [23].
Material properties: BiSbTe, Ref.
[14]; Nao. 9 5Pb 20 SbTe22 , Ref. [31]; PbTe/PbS, Ref. [32]; PbO. 98 T1o-0 2Te,
Ref. [33]; Pbi±,SbyTe, Ref. [34]; n-SiGe, Ref. [16]; p-SiGe, Ref. [17]). .
2-1
22
Experimental (symbols) and computed (curves) thermoelectric properties of n-type bulk Sio.7 Geo. 3 (data from Ref. [51]). Doping concentrations (x10 19 cm- 3 ): o=0.22; A=2.3; 0=7.3; o=17. . . . . . . . . .
2-2
40
Experimental (symbols) and computed (curves) thermoelectric properties of p-type bulk Sio.7 Geo. 3 (data from Ref. [66]). Doping concentration (x101 9 cm- 3 ): E=3.4; A=8.9; o=18; o=24; V=35. x symbols
in the ZT figure indicate the accuracy of the data is questionable as
the ZT value is much higher than previously reported values for bulk
p-type Siij_ G e,.
2-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Experimental (symbols) and calculated (curves) thermoelectric properties of state-of-the-art n-type bulk Sio.Geo. 2. (Solid line - model including carrier concentration variation with temperature; dashed line model without carrier concentration variation with temperature; dotted lines - electronic and lattice components of thermal conductivity.)
42
2-4
Experimental (symbols) and calculated (curve) thermoelectric properties of state-of-the-art p-type bulk Sio.8 Geo.2 . (Solid line - model;
Dotted lines - electronic and lattice components of thermal conductivity.)
2-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Computed results for the electron mean free path versus electron wavelength (left axis), and percent accumulation of electron occupation
number versus electron wavelength (right axis) for n-type (ND ~ 2 x 1020
cm-
3
) bulk Sio. 8Geo. 2 at 300 K. . . . . . . . . . . . . . . . . . . . . .
50
2-6
Schematic of the cylinder model for electrical grain boundary scattering. 51
2-7
Calculated mobility for each scattering mechanism and total mobility
(curves) with experimental data (symbols) for the n-type Sio. 8Geo.2 nanocomposite.
(Solid line - total mobility; broken lines - mobility for each scattering
mechanism .) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-8
Extracted carrier concentration versus temperature for the n-type and
p-type Sio.8 Geo. 2 nanocomposites. . . . . . . . . . . . . . . . . . . . .
2-9
55
56
Calculated (curves) and experimental (symbols) thermoelectric properties for the n-type nanocomposite (data from Ref. [16]). (Solid lines
- model including GB scattering; dashed lines - model without GB
scattering; dotted lines - electronic and lattice components of thermal
conductivity.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2-10 Calculated mobility for each scattering mechanism and total mobility
(curves) with experimental data (symbols) for the p-type nanocomposite. (Solid line - total mobility; broken. lines - mobility for each
scattering mechanism .) . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2-11 Calculated (curves) and experimental (symbols) thermoelectric properties for the p-type nanocomposite (data from Ref. [17]), along with
the bulk experimental data shown earlier for reference.
(Solid lines
- model including GB scattering; dashed lines - model without GB
scattering; dotted lines - electronic and lattice components of thermal
conductivity; dash-dotted line - Seebeck coefficient calculated using the
bulk effective mass of 1.2me.)
3-1
. . . . . . . . . . . . . . . . . . . . . .
59
Thermal conductivity (TC) accumulation (cumulative distribution function) versus MFP. Even though these two distributions could have the
same thermal conductivity, the MFPs of phonons carrying the heat
are completely different. Knowledge of this distribution is critical for
nanoscale engineering.
3-2
. . . . . . . . . . . . . . . . . . . . . . . . . .
66
Phonon dispersions of Al (left) and Si (right). Scattering is permitted
between phonons of the same polarization and frequency. High frequency phonons lacking a corresponding state are diffusely backscattered, indicated by the cutoff mismatch in the figure.
3-3
. . . . . . . . .
73
Lattice surface temperature versus time predicted by the BTE and the
heat equation with truncated relaxation times to 10 ps at 300 K. The
curves are almost identical . . . . . . . . . . . . . . . . . . . . . . . .
3-4
78
Calculated interface conductance G from the BTE with truncated relaxation times to 10 ps at 300 K. The result is close to the specified
value of 1.1 x 108 W/m 2 K, which is chosen at the beginning of the simulation by calculating the transmissivity from Eq. 3.15 using a particular
value of G........................
3-5
........
..
. .. .
79
Lattice surface temperature versus time predicted by the heat equation
and BTE with full relaxation times at 300 K. The two curves do not
match exactly, indicating the transport is in the quasi-ballistic regime.
3-6
80
Calculated interface conductance G from the BTE at 300 K. The interface conductance is not constant as in the diffusive case. . . . . . .
82
3-7
Frequency-dependent 'temperatures' for truncated relaxation times to
10 ps case at 300 K. That the temperatures are almost identical indicates that the phonons have a thermal distribution and are in local
thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-8
83
Frequency-dependent 'temperatures' for the full relaxation time case
at 300 K. The distribution is out of thermal equilibrium, especially at
the interface.
3-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Lattice surface temperature versus time predicted by the heat equation
and BTE with full relaxation times at T = 100 K. Ballistic effects are
more apparent at low temperatures where relaxation times are longer.
85
3-10 Fourier law solutions for k=140 W/mK and k=100 W/mK compared
to the solution of the BTE for an aluminum film on silicon at T =
300 K. Even at room temperature, the BTE solution does not match
that predicted by diffusion theory. Instead, the BTE solution almost
exactly matches a Fourier's law curve with reduced effective thermal
conductivity, indicating quasi-ballistic transport is occurring. . . . . .
87
3-11 Typical multi-pulse responses at different modulation frequencies calculated from the single-pulse response for the BTE and Fourier's law.
Both fits correspond to effective thermal conductivity keff = 100
W/mK and interface conductance G = 1 x 10' W/m 2 K and therefore do not not predict a modulation-frequency dependence of thermal
conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
(a)
Amplitude signal, wo =15 MHz
. . . . . . . . . . . . . . . . . .
90
(b)
Phase signal, wo =15 MHz . . . . . . . . . . . . . . . . . . . . .
90
(c)
Amplitude signal, wo =6 MHz . . . . . . . . . . . . . . . . . . .
90
(d)
Phase signal, wo =6 MHz
90
. . . . . . . . . . . . . . . . . . . . .
4-1
Schematic of the pump-probe system in the Rosenhow Kendall Heat
Transfer Laboratory (from Ref. [96]).
A Ti:Sapphire laser emits a
pulse train of 800 nm, 250 fs pulses at a rate of 80 MHz. The pulse
train is split into pump and probe beams using a 1/2 waveplate and
beamsplitter.
The probe is directed to a mechanical delay line and
subsequently to the sample. The pump is modulated using an electrooptic modulator (EOM), frequency doubled to 400 nm using a Bismuth
Triborate crystal (BIBO), and directed to the sample at the lower left
of the figure. The reflected probe light is send to a fast PIN detector
(amplified or unamplified photodiode); this signal is then sent to a
lock-in am plifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
94
Demonstration of good cable practices in our experiment. Note the
short cables from the photodiode and function generator to the lock-in
am plifier.
4-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
Cylindrical lenses (focal lengths -80 and 130 mm) used to correct the
probe beam astigmatism. . . . . . . . . . . . . . . . . . . . . . . . .
4-4
.
Picture of the curtains used to enclose the experiment and reduce air
currents and dust accumulation. . . . . . . . . . . . . . . . . . . . . .
5-1
99
103
Illustration of the change from the diffusive transport regime (left),
where the heater size D is much larger than MFPs, to the ballistic
regime (right), where the heater size is much smaller than MFPs and
a local thermal equilibrium does not exist. The thermal conductivity
spectroscopy technique measures the change in thermal resistance as
a function of heater size to determine the contributions of different
.
phonon MFPs to the thermal conductivity. . . . . . . . . . . . . . . .
107
5-2
Experimental data (symbols) at T = 90 K for pump beam diameters
D = 60 pm and D = 15 pm, along with the best fit to the a) ampli-
tude and b) phase data from a thermal model (solid lines). To show the
sensitivity of the fit, the model prediction for the best fit thermal conductivity values plus and minus 10% are shown as dashed lines. The
thermal conductivities obtained from the fits are 630 and 480 W/mK,
respectively, different from each other and both far from the accepted
thermal conductivity of 1000 W/mK.o 2. . . . . . . . . . . . . . . . .
5-3
109
(a) Experimental (symbols), calculated by only including the contributions of the indicated MFP or smaller (dashed lines), and the literature
(solid line) thermal conductivity of pure c-Si.10 2 (b) Measured interface
conductance of the Al/Si interface. It is seen that the interface conductance does not depend on beam diameter. (c) Representative measurements of the normalized thermal conductivity versus pump beam
modulation frequency for several temperatures and beam diameters.
No frequency dependence of the thermal conductivity is observed. . . 110
5-4 Experimental measurements (symbols) and first-principles calculations
(lines) of the thermal conductivity accumulation distribution of silicon
versus MFP. The symbols and error bars represent the average and
standard deviation, respectively, of measurements taken at different
locations on different samples, prepared at different times, at four different modulation frequencies from 3-12 MHz. Because of the finite
number of reciprocal space points included in the first-principles calculations, the calculated distribution of MFPs is discrete and cannot
include some long wavelength (and hence long MFP) modes. We use
an extrapolation (dashed lines) to estimate the contribution from these
long wavelength modes . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6-1
Schematic of the metallic nanodot array on a substrate illuminated
by a much larger optical heating beam. Because the effective length
scale for heat transfer is the dot diameter rather than the optical beam
diameter, heat transfer at the nanoscale can be observed, beating the
diffraction limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2
118
Optical image and SEM image of the nanodot array fabricated by
electron-beam lithography (EBL). The numbers in the optical image
correspond to the diameter of the dot (800 = 800 nm, and so on).
Because the MIT EBL machine often exposes a larger area than specified in the layout, a size matrix of several different diameter dots was
exposed for each intended diameter; these different sizes are indicated
. . . . . . . . . . . . .
126
. . . . . . . . . . . . . . .
126
by the number of large dots below each array.
6-3
(a)
Optical image at 10X magnification.
(b)
SEM image at 5 kV, 4000X magnification........
. . ...
126
Typical amplitude experimental data for D=400 nm Al dots on a sapphire substrate. The inset shows data for shorter delay times, where
oscillations in the signal are observed. The oscillations are believed
to be due to a transient diffraction grating effect caused by thermal
expansion and contraction of the dots.
6-4
. . . . . . . . . . . . . . . . .
127
Experimental data (symbols) dot diameters D = 10 pm and D = 400
nm, along with the a) amplitude and b) phase fit to the data from
a thermal model (solid lines) and 10% bounds on the fitted thermal
conductivity value (dashed lines). The thermal conductivities obtained
from the fits are 37 and 22 W/mK, respectively. The value for the 10
pum dots is in good agreement with the literature value for the thermal
conductivity of sapphire, but the value for the 400 nm dots is lower,
indicating that MFPs are in the hundreds of nm in sapphire. . . . . .
6-5
128
Measured thermal conductivity of sapphire versus dot diameter. The
measurements indicate that the dominant heat-carrying phonons in
sapphire have MFPs < 1 pm.
. . . . . . . . . . . . . . . . . . . . . .
129
18
List of Tables
2.1
Fitting parameters for the grain boundary potential used in Sio.8Geo. 2
nanocomposite modeling. . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.1
Parameters used in the phonon relaxation time models for natural silicon. 77
3.2
Values of the effective (keff) and actual (k) thermal conductivities
from the BTE solution at several different temperatures, along with
the MFP cutoff values Ac, required to achieve this thermal conductivity
and the effective penetration depth Lpd at each temperature. The MFP
cutoff values agree reasonably well with the effective penetration depth. 88
A. 1 Lattice parameters for Sii_2Ge. used to calculate the lattice thermal
conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
A.2 Band structure parameters for SiiGe_ used to calculate electrical
properties. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . ..
..
140
20
Chapter 1
Introduction
1.1
Nanoscale heat transfer and thermoelectrics
Nanoscale heat transfer, where energy is transported over scales comparable to the
heat carrier wavelengths and mean free paths (MFPs), is a topic of considerable
interest.1- 7 Transport at these fundamental length scales can deviate strongly from
the predictions of diffusion theory, 8' 9 and materials structured at these length scales
10 1 7 These
have demonstrated properties that are unachievable in bulk materials.
materials have applications in all areas of the energy field, from solar photovoltaics
to thermoelectric waste heat recovery to batteries, to name just a few.
Thermoelectric energy conversion is one technology that has benefited significantly
from the proper application of these nanoscale effects. Thermoelectrics are capable
of converting heat directly into electricity, and can also act as solid state refrigerators or heat pumps which do not use any moving parts or environmentally harmful
fluids.2,18-23 Because of their reliability and simplicity they are used extensively in
24 While
fields such as space power generation and a variety of cooling applications.
their low efficiency compared to that of mechanical cycles presently restricts their
applications, there is a considerable interest in applying newly developed materials to
25
waste heat recovery applications in cars or industry. 24 , This could make a sizeable
contribution to solving the energy challenge as over two-thirds of the energy presently
produced is rejected as waste heat,2" corresponding to almost 26 million barrels of oil
1.51
-
Pb
Pb
Ti2Te
Sb Te
Nano n-SiGe
0PbTe
BiSbTe
0)-
n-SiGe
0.5 -
g'
- 0
0.0
'
0
200
Nano p-SiGe
.
-
p-SiGe
400
600
Temperature (C)
800
1000
Figure 1-1: Figure of merit ZT of current state of the art thermoelectric materials
versus temperature. The dashed lines show the maximum ZT values for bulk state of
the art materials, and the solid lines show recently reported ZT values, many of which
were obtained in bulk nanostructured materials. (From Ref. [23]. Material properties:
BiSbTe, Ref. [14]; Nao.95 Pb 2OSbTe 22 , Ref. [31]; PbTe/PbS, Ref. [32]; Pbo.9 sTlo.0 2 Te,
Ref. [33]; Pbi±.SbyTe, Ref. [34]; n-SiGe, Ref. [16]; p-SiGe, Ref. [17]).
per day. There is also interest in using thermoelectrics for solar thermal applications,
where solar energy is used to create a temperature difference across a thermoelectric
material.2 7 -30
The energy conversion efficiency of a thermoelectric material is determined by
the figure of merit ZT = S 2 .T/k, where S is the Seebeck coefficient, or the voltage
generated per unit temperature difference, o is the electrical conductivity, k is the
thermal conductivity, and T is the absolute temperature at which the properties are
measured.18 '19 To obtain a high energy conversion efficiency, we want ZT as high
as possible, meaning the material should possess a high electrical conductivity, high
Seebeck coefficient, and low thermal conductivity.
These material properties do not often occur in nature. Metals have very high
electrical conductivity but also very high thermal conductivity. Glasses are the opposite, having very low thermal conductivity but also very low electrical conductivity.
The best materials turn out to be heavily doped alloyed semiconductors, which have
low thermal conductivity but electrical properties that can be partially tuned by
doping.1 8 However, the thermoelectric properties are highly interdependent, so that
changing one property results in an adverse change in the other two properties so
that ZT remains constant.
Nanostructuring allows this coupling to be partially broken. In nanostructured
materials, numerous interfaces are able to scatter heat-carrying phonons, or lattice
vibrations, so that the thermal conductivity is reduced significantly while electrical properties are minimally affected. Indeed, remarkably low thermal conductivities in nanostructures such as nanowires, superlattices, and nanocomposites have
recently been reported.13 , 14 3, 5 ,36 Increases in the figure of merit have been reported
12 13
11
10
in some of these materials, including superlattices, quantum dots, nanowires, ,
and nanocomposites.1 4-17 Some of the reports made in recent years are shown in Fig.
1-1. Many different material systems exhibit enhanced figures of merit, and most have
broken the ZT ~1 barrier which was previously the maximum ZT of any material
for over fifty years.
1.2
Outline of the thesis
Despite these advances, much remains unknown concerning the precise mechanisms
by which the heat carriers in thermoelectric materials, electrons and phonons, are
affected by nanostructures.
For electrons, the primary challenge is to understand
how electrically charged defects such as grain boundaries scatter electrons and alter
the electrical conductivity and Seebeck coefficient. While grain boundaries in tech37
nologically important materials such as silicon have been extensively studied, -40 the
magnitude of the grain boundary potential as well as the energy dependence of the
potential are essentially unknown in nanostructured thermoelectrics.
For phonons, the challenges are even greater because we still do not know basic
quantities like phonon MFPs even in bulk materials.
Our basic understanding of
phonon transport is still based on simple relaxation time models which were developed
50 years ago.4,4
The ability to engineer new materials at the nanoscale thus requires
a more complete knowledge of the basic physics of phonon transport.
The purpose of this thesis is to explore the physics of nanoscale transport in
thermoelectric materials and to determine how we can use nanoscale effects to engineer more efficient thermoelectric devices. Chapter 2 studies electron and phonon
transport in nanocomposite thermoelectric materials. We use modeling and materials characterization to understand how carriers are affected by grain boundaries in
materials where the grain sizes are smaller than mean free paths (MFPs). We develop a model to calculate the electrical and thermal transport properties of bulk
and nanocomposite thermoelectric materials using the Boltzmann transport equation
(BTE). By incorporating a new grain boundary scattering model we are able to explain how the thermoelectric properties are affected in nanocomposites as well as to
identify strategies which could lead to more efficient thermoelectric materials.
These types of nanoscale scattering models, while useful, cannot directly provide
information about how heat carriers are affected by nanostructures. The second portion of the thesis focuses on determining how to more directly measure properties of
the heat carriers like the phonon MFP distribution. In chapter 3, we show that in
order to extract this information it is not sufficient to simply measure macroscopic
transport properties; we must study transport at the length scales of the heat carriers
themselves. In the quasi-ballistic regime, where transport occurs over scales comparable to heat carrier MFPs, diffusion theory becomes invalid due to a loss of local
thermal equilibrium. The manner in which the transport deviates from diffusion theory reveals very useful information about the underlying heat carrier distribution. To
explore this concept, in this chapter we numerically solve the frequency-dependent
phonon Boltzmann transport equation, which is valid over length and time scales
smaller than the phonon MFPs and relaxation times, respectively, unlike Fourier's
law. These numerical results confirm that by studying the quasi-ballistic regime, we
should be able to extract information about phonon relaxation times and MFPs.
The rest of the thesis focuses on experimentally measuring MFPs using our ultrafast pump-probe experiment in the Rosenhow-Kendall laboratory. Chapter 4 briefly
discusses the modifications that were made to the system to make it more stable, robust, and automated. These modification greatly simplify the task of data collection.
In chapter 5, we introduce a new technique to measure MFP distributions. By
observing the changes in thermal resistance as a heated area size is systematically
varied, the thermal conductivity contributions from different MFP phonons can be
determined. Using this technique, we measure the MFP distribution of phonons in
silicon for the first time, obtaining good agreement with first-principles calculations.
In chapter 6, we develop a modification of this technique which permits us to study
transport at scales smaller than the diffraction limit of approximately 1 micron, which
is the smallest length scale that can be typically achieved optically. This is important
as many technologically relevant materials like thermoelectrics have MFPs in the deep
submicron regime. To access these length scales, we use electron-beam lithography
to pattern nanoscale metal absorbers with diameters down to tens of nanometers.
By using these nanoscale absorbers as heaters, we are able to investigate phonon
transport in a larger range of materials and length scales. We apply this technique
to measure the MFP distribution in sapphire. The spectrally resolved information
which can be determined using our technique is extremely useful both fundamentally
and for technological applications which involve nanoscale transport.
Finally, chapter 7 examines possibilities for future work and concludes the thesis.
26
Chapter 2
Carrier transport in thermoelectric
nanocomposites
2.1
Background
The first topic we consider is electron and phonon transport in nanocomposites. The
term "nanocomposite" has been used to describe several different types of structures.
The original concept was for either nanoparticles embedded in a host or a heterostructure geometry with nanoparticles of different materials adjacent to each other. 43,4 4
For the heterostructure geometry, when the two materials are the same, the nanocomposite is essentially a material with nanometer sized grains. Because the structures in
the material have dimensions typically in the tens to hundreds of nanometers, length
scales can be smaller than carrier mean free paths, resulting in modified transport
properties compared to those of the bulk material.
Nanocomposite thermoelectric materials, which we also call nanocomposites for
simplicity, have attracted much attention recently due to experimental demonstrations of improved thermoelectric properties over those of the corresponding bulk material. 14 - 17, 3 1, 32 , 4 5, 4 6 These materials have an important advantage over other nanostructured materials in that they can be produced in large quantities and in a form
that is compatible with existing thermoelectric device configurations. The primary
mechanism that leads to ZT improvement is a reduction in lattice thermal conduc-
tivity due to additional phonon scattering at grain boundaries. However, a potential
barrier which forms at the grain boundary (GB) leads to a reduced electrical conductivity but possibly higher Seebeck coefficient. Nanocomposites are commonly created
using either a ball milling and hot pressing technique"-17 or with thermal processing
methods. 31,32 ,45 ,4
6
Sio. 8Geo. 2 nanocomposites with improved thermoelectric properties over those of
the bulk were recently reported. 6 '1 7 To better understand the reported data and to
gain insight into the carrier transport in these materials, we calculate the thermoelectric properties of these materials using the Boltzmann transport equation (BTE).
To perform the calculation for the complex nanostructured material, we split the
computation into two easier parts: first, calculating the properties of the bulk material; second, adding to this calculation a GB scattering rate which accounts for the
additional nanoscale scattering.
For the first task, fortunately the theory for modeling bulk thermoelectric materials using the BTE is well developed and the calculation is straightforward. If the
relaxation time approximation can be used, the thermoelectric properties can be expressed as integrals of a relaxation time;4 7-49 standard expressions for the relaxation
time of various scattering mechanisms are given in the literature. 48-50 For the cases
that we study here the relaxation time approach is a good approximation. A similar
approach was taken by Vining 51 and Slack and Hussain, 52 and we follow their work
closely with a few important exceptions which will be described later.
For the second task, we need to obtain an additional relaxation time
equivalently a scattering rate rG)
TGB
(or
to account for the strong GB scattering mecha-
nism in these materials. Once determined, this scattering rate can be added to the
other scattering rates using Matthiessen's rule and the thermoelectric properties are
calculated in the usual manner.
GB scattering has been of great interest, particularly for device applications involving polycrystalline silicon.
Mayadas and Shatzkes obtained a scattering rate
from a series of delta function potentials. 53 Their model is commonly used to calculate the resistivity of polycrystalline materials. We will show that this model is
not physically consistent for thermoelectric materials. Many other models for grain
boundary scattering in polycrystalline materials attempt to obtain I-V characteristics
by considering thermionic field emission and tunneling processes. 3 7 4 0 However, this
framework neglects the energy relaxation of charge carriers and can result in unrealistically large Seebeck coefficients, especially in nanocomposites. For this reason, we
focus on calculating GB scattering rates, allowing all the properties to be determined
in a self-consistent manner.
The organization of chapter 2 is as follows. First, in Section 2.2, the basic theory of calculating thermoelectric properties using the Boltzmann equation is briefly
discussed. In Section 2.3, the code which implements this calculation is validated by
calculating the thermoelectric properties of bulk Sii_,Ge_ alloys and comparing the
results to experimental data. Section 2.4 introduces models for phonon and electron
GB scattering. We apply the models to Sio. 8 Geo. 2 nanocomposites in Section 2.5 and
compare the results to recently reported experimental data 16 , 17 and new mobility
measurements. Finally, in Section 2.6, we use the model's predictions to help determine which strategies will be most effective in further improving the figure of merit
in these materials.
2.2
Theory
The thermoelectric properties of Sii_2Ge2 can be calculated using the BTE. If all
the scattering mechanisms are elastic, the relaxation time approximation (RTA) may
be employed, thereby simplifying the solution. 4 8- 50' 54 If some scattering mechanisms
are inelastic, then a relaxation time cannot be strictly defined.
For Sii_2Gex, the
only inelastic scattering mechanism is due to optical phonons, where the interaction
energy hwop, wO, being the optical phonon frequency, is on the order of kBT at room
temperature. Fortunately, for practical use of Sii-_Ge_ we are mostly interested in
much higher temperature ranges at which optical phonon scattering can be considered
approximately elastic, and we may use the RTA for our calculations.
A similar approach to calculate the thermoelectric properties of bulk materials
was originally performed by several authors, 51'5 2 and we follow their work closely
but with important exceptions. The common calculations are as follow. First, like
Slack,
2
we assume three band transport, using two conduction bands, one near the
X point and one at the L point; and one effective valence band.
We find the L
conduction band does not contribute substantially to the transport properties. Second, like Vining, 51 we take into account the exact form of the relaxation times from
the literature.4 8 4 9 5 1 5 5 Slack uses an effective relaxation time exponent or empirical
results in his model.
We differ from the authors' calculations in the following ways. The first difference
is that we do not assume that the conduction bands and valence bands have the same
band structure parameters; rather, we use literature values for most key quantities
such as the conduction band effective mass and deformation potential. These values
and any adjustments made are listed in Appendix A.
The second difference is that we take into account the non-parabolicity of the
X conduction band, which strongly affects the thermoelectric properties at high ntype doping concentrations.4 9 Vining and Slack did not take non-parabolicity into
account, and while they were able to fit most of the data, unphysical values of some
fitting parameters were required, or an empirical result was used to obtain agreement.
By using a non-parabolic formulation we are able to explain experimental data over
the entire doping concentration range with only minor adjustments to the literature
values of the band structure parameters.
Another important difference is the treatment of dopant precipitation issues in
SiiGe_. As discussed by many authors,si,52,56-58 Sii_ Ge_ alloys used for thermoelectrics, which are usually doped with P (n-type) or B (p-type), are often doped
beyond the solubility point for the dopant, causing the carrier concentration to vary
with temperature as dopants precipitate out at lower temperatures and become reactivated at higher temperature. These processes can change the carrier concentration
by a factor of two or more over the entire temperature range, significantly affecting the
observed transport properties. This effect is especially pronounced for P in Sii_2Gex,
which has a strong tendency to precipitate at grain boundaries.5 1 Furthermore, in
many cases the carrier concentration depends on the thermal history of the sample,
making it even more difficult to compare results because the properties of the nominally same material at the same temperature might not be equal due to differences
in thermal processing. We have observed this phenomenon in our measurements:
because the mobility is measured using a different system than that used for the reported measurements of electrical conductivity and Seebeck coefficient, the samples
are subjected to different heat treatments and thus the electrical conductivities measured by different systems are not equal. While Vining was forced to fit the chemical
potential at each temperature to account for this variation, we are able to estimate
the change in carrier concentration versus temperature during the electrical conductivity and Seebeck coefficient measurement using the previously reported electrical
conductivity measurements, 16' 17 new mobility measurements, and model calculations.
This procedure simply uses the definition of conductivity o- = nep to estimate n(T).
It is necessary to assume one band transport for this procedure; we find this to be a
valid assumption for the cases studied here. For the n-type case, where the mobility
has a stronger dependence on the carrier concentration, an iterative procedure is used
to ensure self-consistency between all the properties.
A final point to mention before discussing the details of the calculation is the
difficulty in accurately modeling transport in p-type Sii_,Ge, using the Boltzmann
equation. The warped energy surface of the heavy hole band, along with interactions
from the split-off band (which is only 0.044 eV away from the valence band edge),
causes the parabolic energy surface approximation to be poor at high doping levels.
While it is sometimes possible to obtain a reasonable fit to the data, the quality of the
fit is often worse than that of n-type Sii_2Gex, and for the p-type nanocomposite no
satisfactory fit could be found. The valence band parameters used in the calculation
are listed in Appendix A.
2.2.1
Relaxation time approximation
We now discuss the details of the theory necessary to model bulk thermoelectric
materials.
A detailed derivation of these results is given in Ref. [59].
In a bulk
material, under the RTA the electrical current J and heat current
J
carried by
charge carriers can be written as:47
J
=
L
Jq
=
L21 (D
+
dk
+ L22
(
__
L21 J
L12
dT
dx
(2.1)
dx)
(2.2)
d
___
L2_L12L21 )(dT)
L
L2 2
L11J
T
dx
where <D is the electrochemical potential. The coefficients are given by:
L
=
L12
T-
L21
L22
-
F
r(E)V2
2
r(E)v2
(o
T(E)V2
-
r(E)V2
=
(
D(E)dE
(2.3)
) (E - Ef)D(E)dE
(2.4)
(E - Ef)D(E)dE
(2.5)
(E - Ef) 2 D(E)dE
(2.6)
Here E is the energy, D(E) is the density of states, f0 = (exp((E-Ef)/kBT)+I) 1
is the Fermi-Dirac distribution, e is the absolute value of the charge of an electron, v
is the electron group velocity, r(E) is the relaxation time, and Ef is the Fermi level.
Note that it is essential to use the Fermi distribution rather than the Boltzmann distribution as thermoelectrics have a very high doping level and are usually degenerate.
We can identify the electrical conductivity, the Seebeck coefficient, and the electronic
contribution to the thermal conductivity as:
o-
=
dSk =
dT
ke
=
(2.7)
Lu
d4b/dx
dT/dx
L 22 - L12L21
Ln1
_
L12
L(2
(2.9)
We can also calculate the mobility using:
P =-
(2.10)
ne
where n is the electron or hole concentration. Thus, under the RTA, the problem of
calculating thermoelectric properties reduces to calculating the coefficients Lij.
2.2.2
Additional details
To complete the formulation we need expressions for the density of states D(E) and
the electron velocity v, a way to account for multiple band transport, a way to find
the Fermi level, and the relaxation time r. The first two quantities, D(E) and v,
can be obtained from the E(k) dispersion relation. To take into account anisotropic,
non-parabolic bands we use a two band approximation, 49 50 which modifies the E(k)
dispersion relation from the usual parabolic form to:
-(E)=E(1+aE)=-=
2
(L
m*
+2'kt)
m*
(2.11)
where m* and m* are the longitudinal and transverse components of the effective
mass, respectively, and a is the non-parabolicity coefficient, an adjustable parameter
which is approximately a ~l/Eg, with Eg being the bandgap. 4 9 The density of states
can thus be determined from:
D(E)
where m*
-k 2 dk
d
ir2 dE
11/
27r 2
2mD3/2 YI2dy
h2
dE
= N 2 / 3 (m*m*2) 1/ 3 is the density of states effective mass.
(2.12)
Here N is
the number of equivalent valleys, and is equal to six for the X conduction band in
Sii_,Ge_(the electronic structure of SiiGeis the same as that of Si for x < 0.85).60
The electron velocity is:
v
=
1
-VkE
h
=
2 -y1/2
m*
(2.13)
where -y' = dy/dE and m* is the conductivity mass. For a non-parabolic band the
conductivity mass is modified from its usual form as follows:61
f(-9)
3Km*0
m =K4 2K
+ I1 fa tao
(-
(E + aE 2 )3 / 2 D(E)dE
) (E + aE 2 )3 / 2 (1 + 2azE)-1D(E)dE
(2.14)
where K = m*/m* is the ratio of the longitudinal effective mass to the tranverse
effective mass.
Next, we need to account for the presence of multiple bands, including electron and
hole bands. Hole bands have the opposite sign of Seebeck coefficient from electrons
and thus can strongly affect the overall Seebeck coefficent. Multiple band transport
can be incorporated by calculating the contribution from each band and combining
52
the results. The appropriate manner in which to combine the terms is given below:18,
o=
(2.15)
9
y =n
(2.16)
=
(2.17)
S
Z:ki +Z1:'
kelec
.ici: i+ Uj
(S _Sj)2 T
(2.18)
Note the presence of the bipolar thermal conductivity term in Eq. 2.18, which can
be appreciable at high temperatures.
To find the Fermi level we use the charge conservation condition.4 8 ,4 9 The electron
concentration in each band is given by:
ne,i =
j
D(E)(e(E-(E -Ec,)/kBT +
1)dE
(2.19)
where i indexes each conduction band and E is measured relative to the conduction
band edge for each band. The hole concentration in each band is
n 00
ihj =]J
D(E) (e(E(Ej-Ef))kBT + 1 ) 'dE
(2.20)
where
j
indexes each hole band and E is measured relative to the valence band edge
for each band. For charge conservation we need any net electrons or holes to be from
an impurity. This condition gives
NH
-
N'
nhj,
=
e~
-
(2.21)
where ND is the number of ionized donors and N- is the number of ionized acceptors.
NA and N- might not be equal to ND and NA if not all the dopants are ionized, but
since the ionization energy is small for typical dopants such as B and P the number
of ionized donors or acceptors is assumed to equal the number of donor or acceptor
atoms, respectively. 62
Equation 2.21 is an implicit equation for Ef which can be
solved with a numerical scheme such as the bisection method; once Ef is found the
electron and hole concentrations in each band can be determined from Eq. 2.19 and
2.20.
2.2.3
Electron relaxation times
The final quantity needed is the relaxation time, which accounts for the scattering
processes in a material. In this paper we will be modeling Sio. 8 Geo. 2 and will incorporate acoustic phonon (AP) scattering, non-polar optical phonon (NPOP) scattering,
and ionized impurity scattering (IIS). Note that Sii_,Ge, is not a polar material and
thus polar scattering mechanisms are not applicable. Technically Sii-2Gex also has
an alloy scattering mechanism, but our calculations show that alloy scattering is weak
compared to IIS in these highly doped materials and so is not considered here. Standard forms for the AP and IIS relaxation times are in the literature.48-50,54,55,63 The
relaxation times used here incorporate non-parabolicity. The relaxation time rAP for
AP scattering is: 55
rkBTD 2
e-1
T
TAP
(2.22)
"D(E)
pv2h~D
=
E
TO
Eg+2E
I
Dv
DA)
-
_
8j(Di+E)D
3(E, +2E)2 DA
(2.23)
where E is the energy relative to the band edge, Da is the conduction band acoustic
deformation potential, D, is the valence band acoustic deformation potential, Eg is
the band gap, D(E) is the density of states defined in Eq. 2.12, p is the density, and
v, is the sound speed. It is possible to derive other forms for the AP relaxation time
which give similar results.49
The other phonon scattering mechanism, non-polar optical phonon (NPOP) scattering, is not elastic and it is not possible to strictly define a relaxation time. However,
if kBT/hwp > 1, where wo, is the optical phonon frequency, then the scattering can
be considered to be approximately elastic. For NPOP scattering we use an approximation due to Ravich which gives a relaxation time with a form similar to that of
AP scattering.55 The result is:
-1
-1
Tp
-1
=
TO -
2
1 /2 (aD \
rah(kBT)
2
,)0
0
pa2
hwo,
_1 ~
E
Eg+2E
(2.24)
D(E)
1
V
DDo
2
_
8 E(E + E) Dvo
3
(Eg + 2E) 2 Do
where Do is the conduction band optical deformation potential (with units J/m),
D~o is the valence band optical deformation potential, a is the lattice constant, and
the other parameters were defined earlier. The values of the acoustic and optical
deformational potentials are listed in Appendix A.
The final scattering mechanism, ionized impurity scattering, has a relaxation time
48 49
given by the standard Brooks-Herring result: ,
ir (47rZe2/co)
-1_
T
=
8hk4
Fimp = In (1 + ()
=
2
D(E)FimpND
ND
(2.27)
1+
(2kR)2
where Z is the number of charges per impurity,
(2.26)
(2.28)
6o
is the static dielectric constant,
is the doping concentration, k is the magnitude of the electron wavevector, and
R is the Debye screening length. 48 , 49, 61, 64 In the Thomas-Fermi approximation, R is
64
6
given by: 1,
R -2 = 4ire
7e22 Z
&fo )(~D(E)dE
oo"
(0
'E
(2.29)
To obtain a total relaxation time for all the scattering mechanisms, we can add
the scattering rates
-1 using Matthiessen's rule: 48 , 49
T 1
-1
=S
(2.30)
Matthiessen's rule assumes all the scattering mechanisms are independent of each
other. The total relaxation time can then be inserted into Eqs. 2.3-2.6 to calculate
the thermoelectric properties. This completes the theoretical treatment of electron
transport.
2.2.4
Phonon modeling
The final task is to model phonon transport. For this we use Steigmeier and Abeles'
model65 of the lattice thermal conductivity of alloys based on the Callaway model. 41
Their model treats phonons with the Boltzmann equation under the RTA and includes point-defect, phonon-phonon, and phonon-electron scattering mechanisms, all
of which are characterized by a relaxation time as was the case for electrons. 65 These
relaxation times are used exactly as they are given in Ref. [65] and so are not reproduced here.
Once the relaxation times have been computed, the lattice thermal conductivity
can be determined:
k= 2r2 v
kBD
3 (j + 12/13)
(2.31)
where v. is the sound speed, 0 D is the Debye temperature, and 1 ,, 12, and Is are given
by:
I1
=
12
=
/l
]rx
0
2 ax
2
2
(
(exp(arZ)
2
-
1) 2
G22 ax
T
2X 2ea
] j1 -2-o r
(exp(ax)3
37
(2.32)
dx
2
dx
(2.33)
13
Here x
= W/WD,
=
#
II(1
fo
rv (
I OT
u)
2
(2.34)
a(2 1 ) 2 dX
(exp(aX) - 1)2
where WD is the Debye frequency, a = (6D/T)", where
OD
is the
Debye temperature, n ~ 1 is a fitting parameter related to higher order phonon
scattering,
#
is the ratio of Umklapp to normal mode scattering, and T and TU are
the total and umklapp relaxation times, respectively. The total thermal conductivity
is the sum of the lattice thermal conductivity and the electronic thermal conductivity,
Eq. 2.9.
2.2.5
Summary of the calculation
We now review the steps necessary to perform the calculation. After specifying band
structure parameters, temperature, and doping level, the first step is to calculate
the Fermi level. This can be done by calculating ne and nh for each band, as in
Eqns. 2.19 and 2.20, and combining the result in Eq. 2.21. The equation is now a
nonlinear function of the Fermi level Ef. By iteratively solving this equation using
the bisection method, we can find Ef and then the equilibrium electron and hole
concentrations. The next step is to compute the energy dependent relaxation times
and combine them using Matthiessen's rule, Eq. 2.30.
Once the total relaxation
times are determined, we can compute the coefficients Lij given in Eqns. 2.3-2.6 for
each band, and subsequently calculate the properties for each band using Eqns. 2.72.9. The overall transport properties, which are the quantities that are measured
experimentally, are found from combining the properties for each band, as in Eqns.
2.15-2.18. Finally, the lattice thermal conductivity can be calculated separately using
the Callaway model, given in Eqs. 2.31-2.34. This procedure can be repeated for each
temperature or doping concentration, allowing all of the thermoelectric properties to
be determined over the desired temperature and doping concentration range.
2.3
Model validation
To validate the model, we compare the calculated results to experimental data for
bulk Si1isGe. from several sources. 16 , 17 , 51, 66 The band structure parameters used are
listed in Appendix A.
The modeling predictions for bulk n-type Sio. 7Geo. 3, along with the same data
used by Vining for comparison, 51,66 are shown in Fig. 2-1. The calculations match
the experimental data to within about 15% over most of the doping concentration and
temperature range. The two highest doping concentrations have been increased by
about 10% from the reported values; this adjustment is expected to be within the experimental error of the measurement. Some of the important effects discussed earlier
can be clearly seen in the model; for example, for the lowest doping concentration the
bipolar thermal conductivity is dominant at high temperatures, and is accompanied
by a large decrease in Seebeck coefficient.
The most highly doped material shows
evidence of dopant precipitation above 1000 K, seen as an increase in the electrical
conductivity, but this is not accounted for here as the mobility is unknown.
A similar comparison for p-type Sio.rGeo.3 , along with data from Dismukes,66 is
shown in Fig. 2-2. The fit is slightly worse than that for the n-type case, particularly
the Seebeck coefficient, as this parameter is sensitive to the band structure and is
most affected by the failure of the parabolic approximation. At high temperature the
Seebeck coefficient is underestimated by about 20% in the worst case. As indicated
by the x symbols in the figure, the accuracy of some of the experimental data is
questionable as the maximum ZT value for the highest two doping concentrations is
almost ZT = 0.8, higher than the expected value of around 0.5 - 0.6 for state-of-theart bulk Sio. 8 Geo.2 shown in Figure 2-4.
We also computed the thermoelectric properties of state of the art n-type and
p-type Sio.8 Geo. 2 alloys,16' 17 shown in Figures 2-3 and 2-4. In this case the carrier
concentrations are not available, forcing us to adjust the values to fit the data. The
fitted carrier concentrations were determined to be ND = 1.7x 1020 Cm
3
for n-type
and NA = 1.4x 1020 cm- 3 for p-type. The calculation is again in good agreement
-100
-200
-300
0
-400
S-500
-600
21
Temperature (K)
Temperature (K)
1.0
24
7-
0.8
6.
084
0.6
0
0
S
5
U.0.2
3
200
400
800 1000
600
Temperature (K)
1200
200
400
600
800 1000
Temperature (K)
1200
Figure 2-1: Experimental (symbols) and computed (curves) thermoelectric properties
of n-type bulk Sio. 7 Geo. 3 (data from Ref. [51]). Doping concentrations (x 1019 cm-3 ):
o=0.22; A=2.3; E1=7.3; o=17.
E
2500
U
U
S2000
200
o
U
U
300
1500
A
U
0
1000
100
500
U)
200
400
600
800 1000
Temperature (K)
1200
800 1000
600
Temperature (K)
120
X
0.6
V
5
0.4
4
0.2
3'
200
400
0.8
7
6-
200
400
600
800 1000
Temperature (K)
1200
200
U
400
800 1000
600
Temperature (K)
1200
Figure 2-2: Experimental (symbols) and computed (curves) thermoelectric properties
of p-type bulk Sio.7 Geo. 3 (data from Ref. [66]). Doping concentration (x 1019 cm-3):
0=3.4; A=8.9; o=18; o=24; V=35. x symbols in the ZT figure indicate the accuracy
of the data is questionable as the ZT value is much higher than previously reported
values for bulk p-type Sii-,Ge,.
U
Model
- - - No ND(T)
-50
1000
Experiment
-100
800
-150
C
0
-
600
X x,
-200
-250
.2 400
-l
0
£6
-5
400
600
800 1000
Temperature (K)
1200
-300
20
0
400
800
1000
600
Temperature (K)
Total
Xx
4
3
C
2
20 0
e
0.6
e
0.4
xxxxxxxxxxX
Lattice
M- 0.2
Electronic
.
..
.......: ' I -
0
400
1200
600
800 1000
Temperature (K)
1200
200
400
1000
800
600
Temperature (K)
1200
Figure 2-3: Experimental (symbols) and calculated (curves) thermoelectric properties
of state-of-the-art n-type bulk Sio.Geo. 2 . (Solid line - model including carrier concentration variation with temperature; dashed line - model without carrier concentration
variation with temperature; dotted lines - electronic and lattice components of thermal conductivity.)
-
E
1201.
250
1000
1800
oo
o
x
200
-
XK
"xXXX
150
0
600
100
400
-
50
x
w
2uu -
400
200
600
800 1000
Temperature (K)
0200
1200
Model
Experiment
1000
600
800
Temperature (K)
400
1200
U.V
5
Total
0.5
K
4
XXXXXXX
0.4
Lattice
XXXXXXXX
0.3
3
XXXXXXX
0
0.2
2
Electronic
1
0.
200
400
1000
800
600
Temperature (K)
0.1
1200
200
XXXXX
400
800
1000
600
Temperature (K)
1200
Figure 2-4: Experimental (symbols) and calculated (curve) thermoelectric properties
of state-of-the-art p-type bulk Sio.8 Geo. 2. (Solid line - model; Dotted lines - electronic
and lattice components of thermal conductivity.)
with the experimental data over the temperature range. As expected, the n-type
material exhibits a strong carrier concentration variation with temperature due to
dopant precipitation effects. To account for this, the calculated mobility and the
experimental electrical conductivity are used to estimate the carrier concentration
variation with temperature using o- = nep. As the mobility depends on the carrier
concentration to some extent, an iterative procedure to ensure consistency between
all the properties is employed. The carrier concentration variation is found to be
similar to that deduced for the n-type nanocomposite, and using either one will give
a good fit; the fit from the bulk experimental data is used for both Figs. 2-1 and 2-3
as this fit gives a smoother curve.
2.4
Modeling nanocomposites
SiiGe_ nanocomposites can have markedly different transport properties from their
bulk counterparts.
Grain boundaries in materials can act as interfaces to scatter
phonons,6 electrical traps for charge carriers,67-69 and segregation sites for dopant
atoms.51 , 69 In nanocomposites these effects are further enhanced because the grain
size is often comparable to or smaller than characteristic lengths such as the phonon
mean free path or the electron wavelength.
TEM images also show that the mi-
crostructure of nanocomposites is much more complicated than that of microcrystalline materials; in nanocomposites the nanometer-sized grains contain defects and
composition variations, and the nanometer-sized grains themselves are embedded inside larger grains. 1 5- 17 We focus our discussion on the effects of grain boundaries
between the nanometer-sized grains as these are expected to have the largest impact
on transport properties.
The most obvious way the thermoelectric properties are affected by grain boundaries is by a reduction in lattice thermal conductivity, which is the mechanism of
ZT enhancement in Sii_2Ge, nanocomposites. 16
17
It is also found that the presence
of grain boundaries reduces the electrical conductivity and can affect the Seebeck
coefficient. The physical mechanisms for these changes in the transport properties
of polycrystalline materials have been the subject of much discussion, especially for
polycrystalline silicon. 37 , 38 The reduction in lattice thermal conductivity is attributed
to strong interface scattering of phonons.6 For charge carriers, the generally accepted
67
model to explain the effects of grain boundaries is the charge-trapping model. -69
This model postulates that incomplete bonding in grain boundaries leads to the formation of many surface states within the band gap, making it preferable for carriers
to occupy these lower energy states. Due to a depletion of the grain near the grain
boundary, a space charge region with a potential barrier forms; this potential acts as
a scattering potential for charge carriers. Dopant segregation is also thought to affect
the electrical properties of grain boundaries, though if and how the grain boundaries
are affected is not well understood.6 9 Misalignment of crystallographic directions
70
between adjacent grains can also lead to electron scattering processes.
To obtain quantitative predictions of how grain boundaries affect the thermoelectric properties in nanocomposites using the Boltzmann equation, it is necessary to
develop phonon and charge carrier grain boundary scattering models which give a
grain boundary scattering rate -r
The resulting scattering rates can then be added
to the bulk scattering rate, as in Eq. 2.30 for charge carriers, and the thermoelectric
properties calculated in a similar manner to the bulk case. We now develop the grain
boundary scattering models. Here, we specialize the situation for charge carrier scattering to electrons in an n-type material; the same discussion will apply to holes in a
p-type material.
2.4.1
Phonon scattering
There have been several previous efforts to determine the thermal conductivity of
nanocomposites. Yang and co-workers calculated the effective thermal conductivity
of a nanocomposite using the phonon Boltzmann equation.4 3 4 4 Monte Carlo techniques have also been used to calculate the thermal conductivity, giving good results
but requiring significant computational time. 7 1 Prasher has had considerable success
72
obtaining analytical solutions to the Boltzmann equation for simple geometries.
73
Scattering models based on Rayleigh scattering 74 and acoustic Mie scattering theory7 5
have also been used to treat nanoparticle scattering. Minnich and Chen introduced
a modified theory to analytically compute the thermal conductivity of nanocomposites.76
For the present work we use a standard boundary scattering rate. 12,7 4 We can
derive the model by examining characteristic lengths relevant to phonon transport.
In this case the appropriate lengths are the phonon mean free path relative to the
grain size. Our calculations for bulk, poly-crystalline Sio. 8Geo. 2 indicate that a large
fraction of phonons have a mean free path (MFP) APH of about 100 nm, which is
much larger than the nanocomposite grain size
1
g
of about 10-20 nm. A reasonable
model is thus one which approximates that the phonon MFP in nanocomposites is
limited to the grain size such that
APH = 1g.
Then, using the sound speed as the
phonon velocity, as in the Debye model, the scattering rate follows immediately:
-1
TG B - 1
(2.35)
This is added to the other phonon scattering rates, and the lattice thermal conductivity is calculated in the same manner. We are able to obtain a good fit to the
thermal conductivity data using a grain size 1g of ten to twenty nanometers, which is
16 17
consistent with the experimental values recently reported. ,
2.4.2
Electron scattering
The grain boundary model for electrons is based on the charge trapping model, which
postulates that the formation of surface states at the grain boundary depletes the
grain of carriers near the grain boundary, resulting in a space charge region with a
potential barrier. The potential barrier acts as a scattering potential which affects
electron transport. To calculate electrical properties, it is necessary to develop a
model which can describe this phenomenon with a relaxation time, but to create
a consistent model it is necessary to clarify details of the GB scattering process.
Important questions that must be answered are what is a physical value for the GB
potential, whether the electron wave experiences diffuse or coherent scattering, and
over what length scale the grain boundary potential affects the electrons.
The first question, regarding an appropriate value for the GB potential, can be
answered using a depletion region approximation. This analysis yields the following
equation for the GB potential, assuming all the trap states are filled:6 7
U
-
qNt2
8END
(2.36)
where E is the permittivity, ND is the doping concentration, and Nt is the number
density of traps.
For ND
1 X 1020 cm- 3 and Nt = 1 x 1013 cm-2, Ug will be
around 20 meV. Of course, the trap density is unknown and could vary widely. An
important unresolved question is how dopant segregation affects the distribution and
quantity of surface states. 69 It is known that P in Sii_2Ge2 has a strong tendency to
precipitate in the GB," and in principle the extra dopant atoms thus provided to the
GB could affect the surface state distribution. Assuming that the reported values for
N
~ 11X1011 cm-3-1x1013 cm- 3 are a reasonable estimate for our materials, at high
doping levels this model predicts that Ug should be less than 20 meV. 69 However,
we find that a larger GB potential is required to fit the nanocomposite data; we will
discuss possible reasons for this discrepancy in Sec. 2.6.
The last two questions, whether the electron wave experiences diffuse or coherent
scattering and over what length scale the grain boundary potential affects the electrons, can be answered by examining the key length scales related to carrier transport
in nanocomposites as was done for phonons. TEM and XRD measurements indicate
that the average GB size is ten to twenty nanometers and its thickness is around
one nanometer.1 6 '1 7 We can determine how these length scales compare with carrier
length scales by computing the screening length R, the electron wavelength A, and
the electron MFP.
The screening length R was previously given in Eq. 2.29. Computing this value
shows that R is less than one nanometer, implying that the GB potential is completely
screened unless the electron is within a few nanometers of the GB. Hence, we can
conclude that the model should focus on only a small section of the GB, since long-
range potentials from other sections of the GB are neutralized by screening effects.
The Mayadas model53 is not consistent with this result as it models the potential as
a series of grain boundaries.
Next, the type of scattering must be determined. For example, if there are substantial variations in the potential at the GB we might expect the electron wave to be
scattered diffusely, but if the variations are small, the scattering should be coherent.
In addition, if the MFP is comparable to the grain size, we might also expect multiple
scattering effects to be important. These questions can be resolved by calculating the
distribution of electron wavelengths and MFPs. We first determine the cumulative
distribution function of the electron occupation number versus wavelength, which
gives the percentage of electrons that have a wavelength less than a certain value.
The electron wavelength Ae is given by the de Broglie expression:
Ae =
27rh
m*v
27rh(1 + 2aE)
2 mE(1+
-
(2.37)
E
2*(
where m* is the conductivity mass. We define the transport electron occupation
number as:
9t = v2
(Ofo
(2.38)
D(E)
Integrating gt to a certain value and normalizing the result by the integral over all
energy will give the cumulative distribution function for the transport electron occupation number. Additionally, we can determine the electron MFP length using an
equation of the form 1 = vr, where v is the electron velocity and
T
is the total bulk
electron relaxation time. The electron MFP will in general be energy-dependent, and
we can relate the electron MFP to the wavelength Ae by expressing each as a function
of energy.
These two quantities, the transport electron wavelength cumulative distribution
function and the electron MFP versus wavelength, are shown in Fig. 2-5 at 300 K
for heavily doped bulk Sio.8 Geo. 2 . The first observation we can make is that most
electrons have mean free paths between two and five nanometers, which is smaller
than the grain size. This fact implies that each small section of the grain with which
the electron interacts is independent of the others: since the grain size is on average
on the order of ten to twenty nanometers, after the electron scatters from the GB it
will, on average, experience several collisions before it reaches another part of the GB.
Thus the memory of the previous collision is essentially lost by the time the carrier
reaches another part of the grain, and we can conclude that each scattering point
on the GB is independent from the others. This is further evidence against using
the Mayadas model for thermoelectrics as it accounts for the effects of many grain
boundaries scattering coherently.
The second observation is that most electrons have a wavelength between six and
eleven nanometers, which is much larger than the GB thickness of one nanometer.
Diffuse scattering of the electron wave requires substantial potential variation so that
the wave is scattered randomly in all directions. Since the barrier potential is confined
to a region very close to the GB itself, any spatial variation in the potential can
only occur over a length on the order of the GB thickness. Furthermore, having a
large variation in the value of the potential along the GB would require substantial
nonuniformities which have not been observed in microstructure studies. These two
results suggest that any variation in potential is not large and is confined to a region
much smaller than the electron wavelength, which implies that the large variation in
potential required for diffuse electron scattering is not present. Thus we can conclude
that electrons for the most part should scatter coherently from the GB, enabling
the use of scattering theory to calculate scattering rates of an electron wave from a
scattering potential.
The third observation we make is that the electron MFP is predicted to be smaller
than the wavelength, implying that the Boltzmann equation is at the edge of its validity. As the Boltzmann equation still gives good results for both bulk and nanocomposite materials, the above discussion is still expected to hold. This issue is further
discussed in Sec. 2.6.
Based on the above discussion, the physical picture for GB scattering is that of
a carrier interacting with local regions of the GB, with each region acting as an
independent scattering site which coherently scatters an electron wave. The GB is
4.5-
-70
5
4.0 -
-60
Z
E
Z3.5 -
-50
C
ml.
-40
2 3.0 2.5-
-30
2.0-
-20
1.5 -
-10
10
15
Electron wavelength (nm)
1.
28
Figure 2-5: Computed results for the electron mean free path versus electron wavelength (left axis), and percent accumulation of electron occupation number versus
electron wavelength (right axis) for n-type (ND ~ 2x 1020 cm- 3 ) bulk Sio.8 Geo. 2 at
300 K.
composed of many of these scattering sites. The most appropriate model for GB
scattering is therefore one which models only a local potential along a small section
of the GB.
Using these results, we can now create an electron GB scattering model by identifying a scattering potential and calculating the corresponding scattering rate. We
have developed a model which describes a small section of the grain boundary with
a scattering potential Ug in a cylindrical region, as illustrated in Fig. 2-6. Since the
actual GB is an extended line defect, the modeled GB is composed of many such
cylinders, each of which acts as an independent scattering site. Possible variations in
the value of the scattering potential throughout the thermoelectric material are not
considered here. If ro is the radius of the cylinder, z is the direction normal to the
GB, and z = 0 is at the center of the GB, then the model potential is given by:
U9
Uoe-Iz/zo
0
r < ro
(2.39)
r > ro
Here Uo is the maximum grain boundary potential, ro is a constant on the order
of the screening length R, and zo is a constant related to the size of the depletion
region.
This particular form of the potential is chosen for several reasons. The
Figure 2-6: Schematic of the cylinder model for electrical grain boundary scattering.
decaying exponential form of the potential in z is used as an approximation to the
exact potential which would result from a carrier depletion region. The cylindrical
geometry is chosen to model the effects of screening: within the cylinder the GB
potential acts on the electron, but beyond several multiples of the screening length
the GB potential is screened out and is essentially zero.
With the scattering potential determined, the final step is to determine the scattering rate. We use the first Born approximation to calculate the scattering rate
as the GB potential is not expected to be large. The matrix element Mkk, for the
potential is:
Mkk'
--
J
=
47rUo
e FUg(r)e- ik*rd3 r zor
1+ (qzo)2)
J ei'fU,(r')dr
3
2 (Ji(qrro)
0
qrro
}
(2.40)
(2.41)
where Ji(z) is a first order Bessel function of the first kind, q = k - k', and q, and
q, are the r and z components of q, respectively. Here the r - z coordinate system
is relative to that of the disk. From Fig. 2-6, we can express q, and qz in terms of -y,
the angle between the z-axis and k; 6, the angle between the z-axis and k'; and
4,
the angle between k, and k,. The result is:
q, = kz - k' = k(cos - -- cos 6)
(2.42)
q, = k Vsin27+sin26
(2.43)
-
2sin'ysincos#
The scattering rate and momentum relaxation times can be obtained from the
expressions:
S(k, k')
=
h
(2.44)
|2j(Eki - Ek)
|M
(2)3
j
j
S(k, k')(1 - cos O)k' 2 sin Odk'dOd#
j
(2.45)
Since the potential does not have spherical symmetry, evaluating the momentum
relaxation time is more complicated and an analytic solution is not possible. Using
the definitions above, the momentum relaxation time can be shown to be:
8 7r2 U2z20ro4D(E)
0
I(ro,zo)
=
1f
b
2
, j
7o
OJ
o
o
N I (ro, zo)
(2.46)
[sin 6(1 - cos ycos 6 - sin
ysin 6 cos #))
(±(zo 2 2
+ (qzzo)2)2
.(
(2.47)
1ro
dyd~d#
x
(qrro) _
where Ng is the number of cylinders per unit volume. An average has been performed
over the incoming angles y.
This scattering rate has two distinct regimes. For (
r
For
oc Ei
2,
<
1 (low energy electrons),
similar to the energy dependence of a diffusive boundary scattering rate.
> 1 (high energy electrons),
Tj
oc E3/2, similar to the energy dependence of
the ionized impurity scattering rate. Depending on the value of the cylinder radius
ro, this change in energy dependence could give an energy filtering effect.
The final quantity needed is the density of cylinders Ng. If the grain boundary is
modeled as a sphere, the number of cylinders per grain is simply the surface area of
the sphere divided by the base area of the cylinder. The number density of cylinders
is then just the number of cylinders per grain multiplied by the number density of
grains:
N
g
=2
1 4F7r(lgb/2) 2
2
(7rr2)
1
(lgb/
2 )3 xf=(2.48)
x
=gbr2
4f
The factor of 1/2 is necessary as each cylinder is shared between two grains. The
parameter 0 <
f
< 1 is a constant which accounts for the geometrical distribution
of the cylinders. In the actual material, the cylinders are arranged according to the
shape of the grain boundaries, but in the derivation above it is implicitly assumed
that the cylinders are uniformly distributed throughout the material.
This could
lead to an overestimation of the scattering rate. To estimate the magnitude of this
effect, we implemented a simple Monte Carlo simulation which models a particle
traveling through a three-dimensional lattice containing GB scattering sites in various
geometries. In the first case, the GB sites are arranged along the faces of a cube,
an approximation to their actual locations in the material; in the second case, the
same number of GB sites are distributed uniformly throughout the region. After
the particle passes through the GB site it is assumed the particle experiences an
elastic, velocity-randomizing collision; the particle also experiences the same type of
collision over a randomly chosen distance between two and five nanometers to account
for other scattering processes. For each geometry, the number of times the particle
passes through a GB site is recorded. This analysis indicates that assuming the same
number of GB sites are uniformly distributed overestimates the number of times the
particle is scattered by a GB site by approximately 30%. To compensate for this, we
simply set
f
~~0.7-0.8 to reduce the effective density of cylinders. This is valid as it
was previously shown that all the GB scattering sites should be independent of each
other. The results are not particularly sensitive to the value of
f±
f:
using a value of
0.1 will give a GB potential of approximately Ug ± 5 meV. We use
f
= 0.7 for
the calculations in this study.
With the scattering rate determined, the thermoelectric properties of nanocomposites can be determined by adding the GB scattering rate to the other scattering
rates using Matthiessen's rule. To explain the nanocomposite data, suitable values of
the model parameters will need to be determined. The adjustable parameters of the
model are the barrier height U., the radius of the disk ro, and the potential decay
constant zo. However, the parameters are not totally arbitrary: to be consistent with
the characteristic lengths discussed before, ro must be on the order of the screening
length, about one to two nanometers, and zo must be on the order of the space charge
region width, about two to four nanometers. The charge trapping model predicts that
Ug should be less than 20 meV, though the required GB potential turns out to be
larger. In addition, for the lattice thermal conductivity model the grain size must be
specified; this is determined from XRD measurements to be on the order of ten to
16 1 7
twenty nanometers. '
2.5
Nanocomposite results
Using these values as a guide, we have calculated the thermoelectric properties for n16 17
type and p-type Sio.8 Geo.2 nanocomposites whose properties were recently reported. '
The fitting procedure is as follows. First, with all other parameters kept constant, the
parameters ro, zo, and Ug are adjusted so that the calculated mobility agrees with the
experimental results. Next, the carrier concentration versus temperature variation is
determined using the calculated mobility and the reported electrical conductivity. As
mentioned before, we cannot simply use the electrical conductivity from the mobility
measurement because each experimental system subjects the sample to different heat
treatments. Finally, the electrical conductivity, Seebeck coefficient, thermal conductivity, and ZT are calculated. If the calculation is consistent, after fitting the mobility
and carrier concentration the calculated results for all the properties should match
the experimental data.
2.5.1
N-type nanocomposite Sio. 8Geo.2
We are able to obtain excellent agreement with the experimental data for the n-type
nanocomposite using this procedure, indicating that the Boltzmann equation and
Phonon
-----
102
GB
-
S
Total
A
2
4)
10
A
300
400
500
700 800
900
600
Temperature (K)
1000 1100
1200
Figure 2-7: Calculated mobility for each scattering mechanism and total mobility
(curves) with experimental data (symbols) for the n-type Sio.8 Geo.2 nanocomposite.
(Solid line - total mobility; broken lines - mobility for each scattering mechanism.)
the GB scattering model are a good description of the transport. The calculated
and experimental mobilities are shown in Fig. 2-7.
Ionized impurity scattering is
the dominant scattering mechanism over most of the temperature range, with GB
scattering next most dominant at room temperature and acoustic phonon scattering
next most dominant above 1000 K. The GB scattering fitting parameters used to
obtain this result are summarized in Table 2.1.
Figure 2-8 shows the n-type and p-type carrier concentrations versus temperature.
For the n-type nanocomposite, there is a slight decrease in electron concentration with
temperature at intermediate temperature, followed by a large increase in electron
concentration with temperature at elevated temperature. This behavior is consistent
with previous reports, as dopant precipitation to the GB in n-type Sii_2Ge, is known
51 5
to be a significant process even at intermediate temperatures of 600-900 K. ' 8 At
room temperature the material is a supersaturated solution of Sio.8 Geo. 2 and P as the
material has been quenched in air to room temperature, freezing the dopants in place.
As the temperature is increased P is rejected from the lattice as the material attempts
E 3.5-
P-Type
x 3.0-
A.
O
2.50
S2.0-
1.5-
1.,
N-Type
40
600
1 00
800
Temperature (K)
1200
1400
Figure 2-8: Extracted carrier concentration versus temperature for the n-type and
p-type Sio. 8 Geo. 2 nanocomposites.
to return to its equilibrium state. At high temperature, the dopants are reactivated
due to the increasing solubility limit and the electron concentration increases. Somewhat unexpectedly, the presence of additional grain boundaries in the nanocomposite
does not seem to exacerbate this phenomenon; the bulk and nanocomposite materials
exhibit very similar carrier concentration versus temperature curves.
The calculated thermoelectric properties, including dopant precipitation effects,
are shown in Fig. 2-9 (solid curves), along with the nanocomposite experimental data
(symbols) and the calculated results without GB scattering for comparison (dashed
lines). The calculated results which include GB scattering are in excellent agreement
with the experimental results. Dopant precipitation is seen to cause the normally
monotonically decreasing electrical conductivity to actually increase above 1000 K;
the opposite trend is present in the Seebeck coefficient. The same carrier concentration variation with temperature as was used for the bulk n-type case is also used
here.
Table 2.1: Fitting parameters for the grain boundary potential used in Sio.sGeo. 2
nanocomposite modeling.
Material
Ug[meV] ro [nm] zo [nm] t g [nm]
12
2.0
45
1.0
n-type SiO. 8Geo. 2
20
1.0
2.0
45
p-type Sio.8 Geo. 2
E
0
1200
-
-50
-100
%
-
Model
- - -NoGB
0 Experiment
1000
-150
O
0
-600
-- 200
,
200
400
5
-250
000
U
.
600
800
1000
Temperature (K)
1200
71W
00
800
1000
600
400
Temperature (K)
1200
1.5
.
oo
4--
I-
Total
.3
2
S
E 1
0
200
~0.5
attice
.......... ...... .
Electronic
400
800
1000
600
Temperature (K)
~E1.0
1200
-
400
600
800 1000
Temperature (K)
1200
Figure 2-9: Calculated (curves) and experimental (symbols) thermoelectric properties
for the n-type nanocomposite (data from Ref. [16]). (Solid lines - model including
GB scattering; dashed lines - model without GB scattering; dotted lines - electronic
and lattice components of thermal conductivity.)
GB
102 -
GSS
l
E
Phonon
E
XA
10
A
Total
300
400
500
600
700 800 900
Temperature (K)
A
1000
1100 1200
Figure 2-10: Calculated mobility for each scattering mechanism and total mobility
(curves) with experimental data (symbols) for the p-type nanocomposite. (Solid line
- total mobility; broken lines - mobility for each scattering mechanism.)
2.5.2
P-type nanocomposite Sio. 8 Geo.2
Unfortunately, a completely consistent fit for the p-type nanocomposite could not
be found. Specifically, the model is not able to predict the high Seebeck coefficient
that is observed experimentally without making an adjustment to the hole effective
mass. We find that increasing the hole effective mass from 1. 2 me to 1. 5 5 me is able
to explain the data, possibly indicating the nanocomposite valence band is different
from that of the bulk. A change in hole effective mass has been reported in strained
Sii_2Gex/Sii_-Gey thin films,7 7 but whether the same phenomenon is responsible for
the observed nanocomposite properties is not clear.
The mobility is shown in Fig. 2-10. Unlike the n-type case, where ionized impurity
scattering is dominant, here acoustic phonon scattering is the dominant scattering
mechanism over most of the temperature range.
Figure 2-8 shows the hole concentration versus temperature.
Unlike previous
reports for bulk p-type Sio.8 Geo. 2 ,56 the nanocomposite does exhibit a change in hole
concentration at elevated temperature, reducing from around 2.6x 1020 cm- 3 at room
temperature to 2.0x 1020 cm- 3 at 1300 K. This is somewhat unexpected as carrier
concentration changes in p-type Sio. 8 Geo.2 had previously been observed only on the
time scale of thousands of hours. 56 The explanation for this effect is similar to that
300
-1200
E
9
D
3 250
1000
2150
200
3W0001
15
~60100
,-'
g
U
50
2C& 400
600
800
1000
1200
C
00
400
Temperature (K)
5
-
Model
- No GB
--. 1.2 m,
x Bulk Expt
- -
rn
2
--
Nano Expt
600
800
1000
Temperature (K)
1200
-1.00
E
k4
Z3
. . . . . . ...
0.8
---
0.6
Total
U0
2
1
0.
200
0.4
Lattice
,
LL 0.2
Electronic
400
1000
600
800
Temperature (K)
1200
0.0
200
-
400
600
800
1000
Temperature (K)
1200
Figure 2-11: Calculated (curves) and experimental (symbols) thermoelectric properties for the p-type nanocomposite (data from Ref. [17]), along with the bulk experimental data shown earlier for reference. (Solid lines - model including GB scattering;
dashed lines - model without GB scattering; dotted lines - electronic and lattice
components of thermal conductivity; dash-dotted line - Seebeck coefficient calculated
using the bulk effective mass of 1.2me.)
for n-type dopant precipitation. At room temperature, the material is supersaturated
with B, but it must be raised to a higher temperature of about 1000 K to reject B
from the lattice. The B rejection causes the hole concentration to be reduced to a
value closer to the solubility limit.
Figure 2-11 shows calculated thermoelectric properties for different conditions,
along with the experimental thermoelectric properties of the nanocomposite and bulk
material. One set of curves is calculated using an effective mass of 1.5 5 me and includes
GB scattering, while the second set does not include GB scattering but is otherwise
the same. The Seebeck coefficient using the bulk effective mass of 1.2me has also
been computed for comparison.
The calculated Seebeck coefficient obtained using the bulk effective mass of 1.2me
is about 25% lower than the experimental data over the entire temperature range,
while that calculated using 1.55me gives a better fit. An interesting feature of the
experimental data is that even though the hole concentration of the nanocomposite
material is almost twice that of the bulk material (2.6x 1020 cm-3 versus 1.4 x 1020
cm- 3 ), the Seebeck coefficient of the nanocomposite is actually equal to or higher
than that of the bulk. We have tried many different types of GB scattering models,
but none predict the level of increase in Seebeck coefficient that is observed; only
by increasing the effective mass are we able to obtain a satisfactory fit for all the
properties. However, whether the material actually does have a different effective
mass from the bulk is not clear, and more investigation into the transport properties
and band structure of nanocomposite p-type Sio.8 Geo. 2 is necessary.
2.6
Discussion
We used the model calculations to better understand the effects of GB scattering on
the thermoelectric properties. To start, we examine the nanocomposite and bulk experimental data more closely. For both the n-type and p-type materials, the nanocomposite electrical conductivity is reduced from the calculated value without GB scattering at room temperature, but the difference decreases as the temperature increases.
This is expected because the acoustic phonon scattering rate goes as T 3/ 2 and thus
becomes large compared to the other scattering mechanisms at high temperature,
making the influence of GB scattering less significant as the temperature increases.
Crucially, the phonon GB scattering rate is always much larger than the other phonon
scattering rates over the entire temperature range. Thus, while the nanocomposite
electrical conductivity approaches that of the bulk at high temperature, the lattice
thermal conductivity is still much lower than that of the bulk, leading to a net increase in ZT which is most pronounced at high temperature.
Unfortunately, this
also implies that improving the electrical properties at high temperature is difficult
because the mobility is always limited by acoustic phonon scattering, especially in
the p-type material.
The model can provide additional information about GB scattering. An interesting observation is that the GB potentials required to fit the data are several
times higher than those predicted from the Poisson equation.
high trap density of Nt =
ND
=
X1013
Even with a very
cm- 3 and a relatively lower doping concentration
1 X 1020 Cm 3, the magnitude of the GB potential is only predicted to be about
20 meV. However, the value required to fit the data is 45 meV. There could be
several reasons for this. One likely possibility is that there are many more defects
present than were accounted for in the model. TEM images show there are a variety
of nanoprecipitates and composition variations that could also act as electron scattering sites.-15 -1 In the model only scattering sites from the nanoscale grain boundaries
are taken into account, but if the density of scattering sites is actually larger than
this, the magnitude of the GB potential required to fit the data would decrease.
Another possibility is that nanocomposites have more disordered GB regions and
thus more trapping states than have been previously measured for microcrystalline
materials. The nanocomposite materials studied here are doped far beyond the solubility limit for the host material Sio. 8Geo. 2 , with the result that a large fraction of
the inserted dopants are rejected from lattice sites and are forced into other regions
of the material such as the GB. As an example, the p-type nanocomposite sample is
doped with 5% B, or 2.5x 1021 cm-3, but the measured carrier concentration is only
2.6x 1020 cm-3, indicating that most of the dopants do not occupy substitutional
sites in the lattice. It has been previously hypothesized that dopants might affect the
number of trapping states in the GB, though if and to what extent the trapping states
are modified is not known. 69 Unfortunately, it is not easy to determine the number
of trapping states with common methods, such as capacitance measurements, due to
the very small size of the grains.
Yet another possibility is that the grains are becoming so small that the volume
fraction of GB is not negligible, making the material essentially composed of two
phases, one being the host material and the inclusion phase being the GB material.
Since the GB is expected to have very low carrier mobility, the combined material's
mobility will be lower than would be predicted from a scattering model alone.
While it is clear that many questions remain concerning transport in nanocomposites, the results discussed here do suggest several strategies and research topics which
could lead to further improvements in ZT. One obvious topic for further research is
determining how to fabricate nanocomposites with fewer defects and cleaner grain
boundaries. If the number of defects or number of trapping states could be reduced,
the GB scattering rate would be reduced and the mobility would increase, especially
in the room temperature to intermediate temperature range. Understanding the reason for the mobility reduction and adjusting fabrication conditions to minimize it
would thus be very helpful.
Another possible research topic is based on the observation that the lattice thermal conductivity has been reduced so far that it is nearing the electronic thermal
conductivity; in the n-type case the electronic thermal conductivity is actually larger
than the lattice component at high temperature. Unfortunately, this means that further reducing the lattice thermal conductivity will yield only marginal improvements
in ZT as the lattice thermal conductivity approaches the electronic thermal conductivity. Research on ways to reduce the electronic and bipolar thermal conductivities
would thus also be very useful.
The final topic we discuss is the validity of the Boltzmann equation for highly
doped thermoelectric materials. The Boltzmann equation is an equation for particle
transport and neglects wave effects. For this requirement to be satisfied, it can be
shown that the necessary condition is that the electron MFP be much longer than
the electron wavelength.4 8 However, as shown in Fig. 2-5, for highly doped Sio.8 Geo.2
the electron MFP is predicted to be even shorter than its wavelength, making the
applicability of the Boltzmann equation somewhat questionable. The prediction of
the wavelength being longer than the mean free path may also be caused by the
inaccuracy of scattering models, which are mostly derived assuming that the doping
level is not too high. Thermoelectric materials are very highly doped and are usually
degenerate, however, and their electrical properties are sometimes closer to those
of metals than to semiconductors. Thus while the Boltzmann equation has been
successful in explaining most of the experimental data, a predictive capability for the
transport properties of nanocomposites will require a more powerful formalism.
2.7
Conclusion
In this chapter, we have used the BTE under the relaxation time approximation
to investigate the thermoelectric properties of nanocomposite Sii_.Ge2 alloys. We
account for the strong GB scattering mechanism in nanocomposites using phonon
and electron GB scattering models. We find that the calculations are in excellent
agreement with the reported properties for the n-type nanocomposite, but the experimental Seebeck coefficient for the p-type nanocomposite is larger than the model's
prediction. Increasing the hole effective mass gives a better fit, possibly indicating
the valence band in the nanocomposite is different from that of the bulk material.
We also find that dopant precipitation is an important process in both n-type and
p-type nanocomposites, in contrast to bulk SiGe, where dopant precipitation is most
significant only in n-type materials. Finally, the model shows that the grain boundary
potential required to fit the data is several times larger than the value obtained using
the Poisson equation. This suggests that an improvement in electrical properties is
possible by reducing the number of defects in the grain or reducing the number of
electrical trapping states at the grain boundaries.
To conclude this chapter, using modeling along with experimental characterization
of our nanostructured materials has given us very useful insight into the nanoscale
transport processes taking place in these materials. These kinds of relatively simple
computations allow us to gain a much deeper understanding of nanoscale transport
than if we only examine experimental data.
64
Chapter 3
Determining phonon mean free
paths: the quasi-ballistic heat
transfer regime
3.1
Background
In chapter 2 we saw that modeling, combined with experimental characterization, can
give us useful insight into nanoscale transport processes. However, one problem with
this approach is that it is indirect: while the model may be able to fit the data, there
is really no guarantee the model is correct. Even completely unphysical models can
fit the data presented in chapter 2 if the numbers are tweaked appropriately.
Because of this dilemma, it is natural to wonder if there is a way we can more
directly study the heat carriers themselves, rather than relying on indirect models.
Unfortunately, if we restrict ourselves to only measuring macroscopic transport properties, like was done in chapter 2, we quickly run into a problem.
Let us examine the kinetic equation for the phonon thermal conductivity. We can
express the thermal conductivity as:
k
=
-
3 10
Cov, Adw
(3.1)
N
E
0
0.8
Nanostructuring effective
/
,0 0.6
|-0.4
E 0.2
Nanostructuring less effective!
-2
102
10
00
10
MFP (pm)
Figure 3-1: Thermal conductivity (TC) accumulation (cumulative distribution function) versus MFP. Even though these two distributions could have the same thermal
conductivity, the MFPs of phonons carrying the heat are completely different. Knowledge of this distribution is critical for nanoscale engineering.
where C, is the specific heat, v, is the group velocity, A, the mean free path (MFP),
wm
is the cutoff frequency, and w the phonon frequency. From this integral, we see
that if we only measure macroscopic transport properties like the thermal conductivity, we measure the integral effect of all phonons contributing to the heat transfer.
The information about the underlying distribution is not easily accessible. In fact,
any distribution of MFPs A, can give the thermal conductivity we measure, if the
numbers are adjusted appropriately. Therefore, simply measuring macroscopic properties cannot reveal properties of the distribution. To directly measure properties of
the carrier distributions, like MFPs A, or relaxation times
T
= A/v,, we need a
new strategy.
Before discussing what the strategy is, it is necessary to emphasize why this information is so important. The reason is illustrated in Fig. 3-1, which shows the
thermal conductivity accumulation, or cumulative distribution function (CDF) of the
thermal conductivity, as a function of the MFP. This shows how much different MFPs
contribute to the total thermal conductivity. In principle, both of these distributions
could have the same thermal conductivity after the distribution is integrated. However, the two distributions are completely different: it is difficult to engineer the
thermal properties of materials with the blue distribution because the MFPs are already very short. Meanwhile, the opposite is true for the green distribution: it is
much easier to engineer the thermal properties in this material because there are
many long MFP phonons which can be modified by nanostructuring. Unfortunately,
this dichotomy cannot be revealed by macroscale transport measurements.
Clearly, our lack of knowledge concerning MFPs is of concern, because it is very
difficult to engineer a material at the scales of the MFPs unless we know their values.
While nanostructured materials such as nanowires, superlattices, and nanocomposites with strongly reduced thermal conductivities have recently been reported and
are being assessed for use in thermoelectric applications, 13,14,
35 , 36
further engineering
these materials will require a better understanding of phonon MFPs.
3.2
The quasi-ballistic heat transfer regime
In order to directly measure properties like MFPs, we need to study transport at the
fundamental scales of the phonons themselves. Consider a typical macroscale thermal
measurement. If thermal transport is studied in the diffusive regime, or for length and
time scales much longer than MFPs and relaxation times, the phonons have already
relaxed to a near-equilibrium distribution. Because of this, the relaxation information
is not easily accessible.
In the quasi-ballistic regime, however, where length and time scales are shorter
than MFPs and relaxation times, the phonons have not had a chance to relax due to
a lack of scattering events. As a result, local thermal equilibrium does not exist and
the transport is non-diffusive. By observing transport at these scales and watching
the phonons relax, we can extract information about MFPs or relaxation times.
This type of size effect, where there is a large temperature gradient over the scale
of phonon MFPs, has been studied both theoretically and experimentally. Transient
ballistic transport has been studied using heat-pulse techniques at cryogenic temperatures in low defect materials such as single-crystal sapphire, where phonon MFPs
are comparable to the dimensions of the sample itself.78 A nonlocal theory of heat
transport was proposed as a modification of diffusion theory in the case where some
phonon modes are ballistic.9 Chen predicted that the heat conduction from a heat
source smaller than the phonon MFP of the surrounding medium is significantly reduced.8 Such non-diffusive transport has important implications for heat transfer in
microelectronic devices. 79
As an introduction to this transport regime, in this chapter we numerically solve
the frequency-dependent Boltzmann transport equation (BTE), which is valid at
length and time scales shorter than MFPs and relaxation times, to better understand
how the quasi-ballistic transport regime can yield information about the phonon distribution. We solve the BTE for a metal film on a substrate using the method of
discrete ordinates. The metal film on a substrate geometry is the same as that used
in transient thermoreflectance (TTR) experiments.80 In subsequent chapters we will
experimentally demonstrate a technique which uses observations of quasi-ballistic
transport to extract information about phonon MFPs.
There have been many numerical solutions to the phonon Boltzmann transport
equation (BTE) in past years, 7' 81-85 but many studies used frequency averaged properties. In a material such as Si, where phonon MFPs can vary by five orders of
magnitude over the Brillouin zone,86 this assumption is clearly not realistic. While
some studies did include frequency dependence, 81-83,85 several of these studies made
other simplifying assumptions which are not self-consistent .81,82 A calculation of the
phonon transport through a single material, including dispersion and polarization,
has been reported, but the calculation did not include interfacial transport.8 3
Here we include the frequency-dependence of the phonon properties in both the
metal film and the substrate, allowing for a frequency-dependent interface condition.
We discuss the definition of thermal boundary resistance at the metal-substrate interface including frequency dependence. We use the model to show that Fourier's
law does not correctly describe the transport in the quasi-ballistic case, and to examine how observing quasi-ballistic transport can reveal information about the phonon
relaxation times.
3.3
Numerical solution of the frequency-dependent
phonon Boltzmann transport equation
Here we describe our numerical solution of the transient, one-dimensional, frequencydependent phonon BTE. To simplify the analysis, heat conduction due to electrons
in the metal film is neglected; only phonon heat conduction is considered. In reality,
of course, electrons conduct a majority of the heat in metals. However, it has been
shown that a thermal resistance due to electron-phonon coupling exists which modifies
the effective interface conductance value. 87 As we are interested in observing changes
in thermal properties due to quasi-ballistic transport only, to simplify the interpretation we do not include electron heat conduction in the calculation. The thermal
conductivity of the metal film is taken to be only that of the lattice, approximately
20 W/mK.
The BTE is given by:81
+Ofepos afefW
-W
at
Here
fh =
ax
-
fo
(3.2)
TW
wD(w)g(w)/47r is the desired distribution function, where h is Planck's
constant divided by 27r, w is the angular frequency, D (w) is the phonon density of
states, and g(w) is the occupation function;
f2 is the equilibrium distribution function,
p = cos 0 is the cosine of the angle between the propagation direction and the x-axis,
and -r is the frequency dependent relaxation time. The factors of 47r normalize the
distribution by the solid angle. Other treatments of the BTEsi' 8 2 have written the
equation in terms of the intensity 1 =
fvow;
however, as will be shown later, optical
phonons must be included in the calculation but have essentially zero group velocity.
To allow for the case of zero group velocity, we remove the velocity from the definition.
The total thermal conductivity of metals is often in the hundreds of W/mK range,
but as discussed earlier, much of this is due to electrons. Here we only consider the
conduction due to the lattice. The lattice thermal conductivity of the metal is not
known with certainty; we use a value of k, ~ 20 W/mK. Phonon relaxation times are
also not well known in metals, and we take the relaxation time T to be a constant so
that k = 1/3
f Covrdw
gives the specified thermal conductivity.
The equilibrium distribution f0 is given by
0
(33)
hwD (w) fBE(T)
41r
where
fBE(T)
= (exp(hw/kBT) - 1)-1 is the Bose-Einstein distribution function, and
T is the temperature of the distribution.
For this study, we will take the materials to be an Al film on a Si substrate
because these materials are commonly studied using TTR. This equation can be solved
numerically for each frequency in both the film and the substrate, an appropriate
interface condition applied, and the solution obtained. We now discuss several issues
which must be treated properly to obtain a self-consistent solution.
3.3.1
Calculation of the equilibrium distribution
The equilibrium distribution
fo
can be determined by integrating equation 3.2 with
respect to frequency and angle and invoking conservation of energy.81 The result is:
j j dd
=
(3.4)
d pdw
We now have a choice about how to satisfy this equation. In the past, the equation
has been satisfied by enforcing equality at each frequency, or
f0
= 1/2
f' f
dy.
However, it has been shown that this approximation will not give the correct solution
to the BTE.85
Therefore, it is necessary to determine the temperature T by integrating Eq.
3.4 over frequency and angle. To simplify the calculation, we approximate that the
temperature difference throughout the domain is not too large. We can then linearize
the equilibrium distribution:
f
where AT = T - To.
~~f2(To)+
((T)
=
_
47r
wAT
47ir
(3.5)
This gives a simpler equation for the temperature of the
equilibrium distribution:
AT =
4r
fomm(Cw/rQ do
jrj
f- f0(To)ddpd
"
o0
-
_1
(3.6)
7rW
The equilibrium distribution then follows immediately from equation 3.5.
Unfortunately, solving for the equilibrium distribution in this manner significantly
complicates the solution. Since all the frequencies are coupled, Eq. 3.6 must be solved
at each spatial point. In silicon, phonon MFPs can vary from a few nanometers for
zone-edge phonons to millimeters for long wavelength phonons, a six order of magnitude difference. To obtain a correct solution for short mean free path phonons,
the numerical spatial grid must have a small step size. However, to obtain a correct solution for the fast, long MFP phonons requires a large spatial domain. Since
the solution must be known at every spatial point, these contradictory requirements
imply a large domain with very fine spatial step size is required, making the solution computationally demanding. Multi-grid schemes, with a different spatial grid for
each frequency, could be used, but to simplify numerical considerations we solve the
distribution function for all frequencies on a single grid.
3.3.2
Role of optical phonons
Optical phonons are typically neglected in studies of phonon transport, as they contribute little to the thermal conductivity due to their near zero group velocity.8 6
However, in transient heat transport, optical phonon modes with large specific heats
act as a thermal capacitance and thus affect the transport, even if the modes do not
actually transport any heat themselves.83 8,
We incorporate these phonons in Si (Al
does not have optical phonons) simply as another mode with zero group velocity. We
must also specify an optical phonon relaxation time; we use a value of 3 ps. 89 The
solution is not sensitive to the precise value. The optical phonon specific heat is given
by the Einstein model assuming three degenerate optical branches:4 7
S3NkB
ChOPp
kBT
O
2
exp(hwop/kBT)
(exp(hwop/kBT) _ 1)2
where N is the number density of the material and wo, is the optical phonon frequency,
equal to 63 meV in Si. 6 0
3.3.3
Interface condition
After the phonons travel through the metal film they reach the interface between
the metal and substrate, where phonons are either transmitted or reflected. Unfortunately, the details of these transmission or reflection processes are largely unknown. A
phonon could be scattered diffusely, or randomly in all angles; or specularly, in a mirror type reflection. Phonons could also scatter inelastically, resulting in a change in
phonon frequency, or undergo mode conversion, resulting in a change in polarization.
In our treatment of the interface, we assume elastic, diffuse scattering and neglect
any type of mode conversion: phonons do not change frequency or polarization as
they cross the interface, but are scattered equally in all directions. A diagram with
the dispersions of Al and Si, along with the allowed scattering modes, is shown in
Fig. 3-2.
We can split the distribution function into forward-going (0 < p < 1) and
backward-going (-1 < p < 0) phonons at both sides of the interface. Generally the
incoming fluxes to the interface, or the forward going flux on side 1,
fi, and
back-
ward going flux on side 2, f-2, are known; the outgoing fluxes, or backward going flux
on side 1,
f-1 , and
the forward going flux on side 2,
fl2 , need to
be determined.
The interface condition simply expresses that the heat flux carried by
equal to the reflected and transmitted heat flux from
fi and f-2 .
f
1
and
f,2 is
Because we assume
elastic scattering and neglect mode conversion, the heat flux equality condition must
Cutoff mismatch
AIls Cr
Si
U-
Cutoff mismatch
Frequency and branch match
r
X
Wavevector
Wavevector
Figure 3-2: Phonon dispersions of Al (left) and Si (right). Scattering is permitted
between phonons of the same polarization and frequency. High frequency phonons
lacking a corresponding state are diffusely backscattered, indicated by the cutoff mismatch in the figure.
be satisfied for each frequency and polarization. The condition is:
q+=
jfiv
qj
=
jf-
fo2 Vw2
f,-ivi
pdy
(3.8)
pdy
(3.9)
=
2 (T12 (w)q+ - R 21 (w)q2)
(3.10)
=
2
/10O
2v 2
(-T21(w)q- + R12(w)q+)
(3.11)
Here T 1 2 (w) is the transmission coefficient from side 1 to 2, R 2 1 (w) is the reflection
coefficient from side 2 back into side 2, and so on.
Under the assumption of elastic scattering and the neglect of mode conversion,
some high frequency phonons are unable to transmit from one material to the other
due to a lack of a state. For Al and Si, this occurs for phonons in the Al transverse
branch and the Si longitudinal branch. These phonons are not able to transmit and
therefore have zero transmissivity; this is indicated by 'cutoff mismatch' in Fig. 3-2.
Previous analyses did not consider cutoff frequency mismatches between two materials
or frequency dependent transmission and reflection coefficients, and it is necessary to
reexamine the relations between the coefficients to ensure that frequency dependence
is properly incorporated.
There are several restrictions on the values of the transmission and reflection
coefficients so that the principle of detailed balance and conservation of energy are
satisfied. Let us assume the frequency-dependent transmissivity from medium 1 to
medium 2, T 12 (w), is specified. The principle of detailed balance requires that when
both materials are at an equilibrium temperature T, no net heat flux can transmit
47
across the interface. Typically this condition is written as an integral over frequency,
but it can be shown that the principle of detailed balance applies for every phonon
mode on each side of the interface.9 0 The condition is:
T12(W)fi(T)vei = T2 1(w)f. 2 (T)vo2
(3.12)
Thus, even if T 12 (w) is frequency independent, T 21 (w) will in general be frequency
dependent. Because we have assumed diffuse scattering, none of the transmission or
reflection coefficients are angle-dependent and no further restriction is necessary.
Note that because f,(T) depends nonlinearly on T, temperature cannot be extracted from Eq. 3.12, meaning that the relationship between T 12 (w) and T2 1 (w)
required to satisfy detailed balance will change depending on the phonon temperature. However, due to the small deviations from the equilibrium temperature To
considered here, to excellent approximation detailed balance can be satisfied for all
the phonons by evaluating Eq. 3.12 at To.
The rest of the relations between the transmission and reflection coefficients can
47
be derived by enforcing an equality of the heat fluxes on each side of the interface,
giving:
R 12 (w)
=
1-T
(w)
(3.13)
R 2 1 (w)
=
1 - T 21 (w)
(3.14)
12
All of these conditions must be satisfied at every frequency and polarization.
Finally, the transmissivity T 12 (W) can be related to the interface conductance G
by calculating the heat flux and equivalent equilibrium temperature on each side of
the interface. 7 The interface conductance is defined as G = q/AT, where q is the
heat flux at the interface and AT the temperature difference across the interface.
The heat flux depends on the transmissivity, and therefore will be different from the
results derived previously7 due to the frequency dependence of the transmissivity.
The temperature difference AT is difficult to define consistently. Assuming that
fji
and
f,-
will be composed of reflected and transmitted phonons at these different tempera-
f32
have some emitted phonon temperature Tei and Te2 , respectively,
fj2
and
tures and will be strongly out of equilibrium. 7 It has been shown that the appropriate
quantity to use is the equivalent equilibrium temperature, which represents the temperature of the distribution that a non-equilibrium phonon distribution would reach
if it were to relax adiabatically to equilibrium.7 Thus, even though the forward and
backward going phonons are strongly out of equilibrium, using the equivalent equilibrium temperature allows the results to be compared to the Fourier's law result which
assumes local thermal equilibrium.
Using these results, the modified relation between the interface conductance and
transmissivity is:
(T12(U))f,'osi)
=2_
(Cv)- 1 -
(3.15)
(Cv)21 + (2G)-1
where (-) denotes integration over frequency. The details of the derivation are presented in appendix B. If the transmissivity is frequency independent then this formula
reduces to the result derived previously. 7 It is important to remember that this formula assumes that the incoming distributions to the interface,
fji
and
f-2,
have a
temperature. If these distributions are also out of equilibrium, then it is not possible to relate the interface conductance and transmissivity. In this case, with the
transmissivity specified, the interface conductance will change with time until the two
incoming distributions do have a temperature, at which point Eq. 3.15 will be valid
and the interface conductance will become a constant. Due to the transient nature of
the heat transport studied here, the two distributions never exactly achieve a thermal
distribution, and so the thermal conductance can still change slowly with time.
Phonon dispersion and relaxation times
3.3.4
We now need to specify the dispersion and relaxation times for both the Al film and
the Si substrate. We use the experimental dispersion in the [100] direction for both
Al and Si.
We assume a constant relaxation time T for all modes in Al; the value T
=
10 ps is
chosen to yield the desired lattice thermal conductivity k ~ 20 W/mK. For Si, we use
relaxation times for phonon-phonon scattering extracted from molecular dynamics
simulations9 1 but with an empirical term exp(-9/T) to extend the relaxation times
to lower temperatures.42 We also add boundary scattering and impurity/isotope scattering, important at T < 100 K, and combine the relaxation times using Matthiessen's
rule -
1
= E
T-
1
.
The relaxation times are:
TL
1
=
ALW 2 T1.49 exp(-/T)
(3.16)
Ti
1
=
ATW 2 T 1 .65 exp(-O/T)
(3.17)
ri 1 =
Ajw4
(3.18)
r1
Wb
(3.19)
=
where L and T denote longitudinal and transverse, respectively. The parameters used
are specified in table 3.1.
The thermal conductivity is calculated with the above relaxation times using the
kinetic theory expression for the thermal conductivity:
k =
CWV2
TdW
(3.20)
where the sum is over polarization p. The exact dispersion is used in this calculation.
As in Holland's model, 42 the additive term accounting for normal scattering from
Table 3.1: Parameters used in the phonon relaxation time models for natural silicon.
Parameter
Value
AL [K. 4 9 /s]
2 x 10-19
[K1. 6 5 /S]
1.2 x 10-'9
AT
0
[K]
80
[S3]
x 10-45
A,
3
Wb [S-1
1.2 x 106
Callaway's model is neglected.'
3.3.5
Numerical details and boundary conditions
We solve the BTE numerically using a discrete ordinates method.8 2 Both the angle
and frequency are discretized using Gaussian quadrature to minimize the number
of points required.
The angle is discretized into N,, = 40 points. The frequency
discretization is more complicated because of the cutoff frequency mismatch for zoneedge phonons. Integrals involving the phonon dispersion are split into two integrals,
one over the modes in common between Al and Si and another over the remaining
modes, which in this case is the high frequency modes of the Al T branch and the Si
L branch. These integrals are then separately discretized using Gaussian quadrature.
An explicit first-order finite difference method is used to discretize the spatial and
temporal derivatives. 82
In addition to the interface conditions, one boundary condition is required for each
angle -1 < p < 1. The distribution function must be specified for angles 0 < p < 1
at x = 0, or the top of the metal film, and for angles -1 < p < 0 at x = L, or
the bottom of the substrate, where L is the length of the numerical domain. We
choose the boundary condition at x = 0 to be diffuse reflection so that this surface
is adiabatic. For the boundary condition at x = L, the type of boundary condition
used ideally should not matter if the domain is sufficiently long. We use adiabatic or
constant temperature boundary conditions and verify that the solution ot the BTE
is the same in both cases.
The initial condition is an exponentially decaying phonon temperature distribution, with the 1/e depth of the temperature profile taken to be the light absorption
depth in Al, approximately 7 nm for visible wavelengths. 9 2
0.12.-
E 0.08 0.06-
0.040.020
2000
6000
4000
Time (ps)
8000
10000
Figure 3-3: Lattice surface temperature versus time predicted by the BTE and the
heat equation with truncated relaxation times to 10 ps at 300 K. The curves are
almost identical.
The interface transmissivity T1 2 (w) is calculated using Eq. 3.15 for a particular
value of G specified at the beginning of the simulation. The rest of the transmission
and reflection coefficients can then be calculated as described in section 3.3.3.
3.4
Results
We first consider a test case. In this simulation, performed at T=300 K, the maximum
relaxation time was truncated to 10 ps, putting the transport easily into the diffusive
limit. In this case, the solution from the BTE should be very close to the Fourier
law solution. Fig. 3-5 shows the surface temperature of the metal film versus time
predicted by the BTE and by the heat equation. The length of the domain is 3 Pm.
To verify that the finite length of the domain does not influence the solution, two
different boundary conditions, adiabatic and constant temperature, were employed
at the edge of the substrate, and the length of the domain was varied. In all cases
1.25-
-
1.2-
1.15
0
1Z'
o1.15
10
2000
40 100
60100
Time (ps)
80100
10000
Figure 3-4: Calculated interface conductance G from the BTE with truncated relaxation times to 10 ps at 300 K. The result is close to the specified value of 1.1x108
W/m2K, which is chosen at the beginning of the simulation by calculating the transmissivity from Eq. 3.15 using a particular value of G.
the solutions were nearly identical, indicating that the spatial domain is sufficently
large. To verify convergence of the numerical solution, the number of spatial points,
angle points, and frequency points were all increased; the solution again remained
identical.
We can also calculate the interface conductance and compare it to the value specified at the beginning of the simulation. The result is shown in Fig. 3-4; the calculated
result is close to the specified value of 1.1 X10ol W/M2K
This test case demonstrates that the calculation successfully reproduces the diffusive limit and is operating properly. We now perform the same simulation, only with
the restriction on relaxation times removed; relaxation times are now given by the
equations in Sec. 3.3.4 and can be arbitrarily long. The same figures as before, lattice
surface temperature versus time and interface conductance versus time, are shown in
Figs. 3-5 and 3-6, respectively.
0.12*d0.1 0.18
E
I0.06-
0.040.02
0'
0
2000
6000
4000
Time (ps)
8000
10000
Figure 3-5: Lattice surface temperature versus time predicted by the heat equation
and BTE with full relaxation times at 300 K. The two curves do not match exactly,
indicating the transport is in the quasi-ballistic regime.
This calculation gives an unexpected result. Examining Fig. 3-5, we see that at
room temperature the solution from the BTE does not match that from diffusion
theory, indicating ballistic effects are present. This is somewhat surprising, considering that estimates of the average phonon relaxation times are around 50 ps in Si. 4 7
This illustrates the danger of using the concept of an average relaxation time in
many materials. As mentioned in Sec. 3.2, despite the average relaxation time being
relatively short, more detailed analyses of phonon transport reveal that MFPs and
relaxation times can vary by 5-6 orders of magnitude over the Brillouin zone. In Si,
it is estimated that 40% of the thermal conductivity is contributed by phonons with
MFP longer than 1 pm.9 1 In terms of the present one-dimensional simulation, these
long MFP (and relaxation time) phonon modes do not scatter over the timescale of
the simulation, resulting in a non-equilibrium phonon distribution and the failure of
Fourier's law, which assumes local thermal equilibrium. From Fig. 3-5, we see that
the surface temperature decay curve is shallower than the Fourier law prediction, indicating that the heat transfer in the quasi-ballistic case is smaller than in the diffusive
case. This phenomenon of "ballistic thermal resistance" in quasi-ballistic transport
has been demonstrated experimentally in sapphire,9 3 in semiconductor alloys,9 4 and
in the subsequent chapters of this thesis.
Examining the interface conductance in Fig. 3-6, we see an additional unexpected
result.
Instead of a constant interface conductance close to the value of 1.1x108
W/m 2 K specified, the interface conductance changes with time. The reason for this
was discussed in Sec. 3.3.3. To define an interface conductance, the incoming distributions at the interface must have a thermal distribution. As shown earlier, due to
the long relaxation times of some phonons, a local thermal equilibrium does not exist
over the short timescales of this simulation, making it impossible to define a constant
interface conductance.
To better understand this effect, let us examine the angle-integrated intensity for
each frequency normalized to the specific heat:
T
f
fedpi
CW
= f,(3.21)
E
1.25-
-o
X
M
t0
.
1.1-
1.05-
10
2000
6000
4000
Time (ps)
8000
10000
Figure 3-6: Calculated interface conductance G from the BTE at 300 K. The interface
conductance is not constant as in the diffusive case.
This gives a 'temperature' for each frequency. If the distribution is in thermal equilibrium this temperature will be constant for all frequencies. This quantity is plotted
at t=10 ns for the 10 ps truncated and full relaxation times. As shown in Figs. 3-7
and 3-8, the 10 ps case has a well defined temperature even at the interface, while the
full relaxation time case does not. This indicates that the phonon distribution is not
in thermal equilibrium for the full relaxation time case, as expected. Therefore, the
assumption made in the derivation of the relation between thermal conductance and
transmissivity is not strictly valid, and it is not possible to define a constant interface
conductance.
These ballistic effects are much more apparent at T< 300 K, where phonon relaxation times can be substantially longer than 1 ps. A typical result is shown in Fig.
3-9 for T=100 K. There is a much more marked difference between the heat equation
and BTE solutions, indicating that the transport is quasi-ballistic and a local thermal
equilibrium does not exist.
:10-3
2-
2
1.5(U
E1
a15
0.1
-105
0.15
0.5-
0'
0
0.5
1
1.5
x
2
2.5
3
([Lm)
Figure 3-7: Frequency-dependent 'temperatures' for truncated relaxation times to
10 ps case at 300 K. That the temperatures are almost identical indicates that the
phonons have a thermal distribution and are in local thermal equilibrium.
a-
n
0.5
1
1.5
2
2.5
3
x (sm)
Figure 3-8: Frequency-dependent 'temperatures' for the full relaxation time case at
300 K. The distribution is out of thermal equilibrium, especially at the interface.
3.5
Discussion
From the results presented, it is clear that qualitatively, the solution of the phonon
BTE compared to the Fourier's law solution contains information about relaxation
times. In particular, if the BTE solution is different than the Fourier's law solution, we
can conclude that some phonon modes have relaxation time longer than the timescale
of the simulation. The magnitude of the difference between the two solutions also
gives some information about relaxation times: the longer the relaxation times, the
larger will be the deviation and vice versa.
We would like to better understand which phonon modes are responsible for these
discrepancies and determine how to extract information about relaxation times. As
discussed in the previous section, quasi-ballistic transport results in a smaller heat
flux than predicted by Fourier's law, corresponding to a smaller effective thermal
conductivity. We can interpret the value of this effective thermal conductivity using
00.12a
E
0.1-
0.
E 0.08 0.060.040.020'
0
2000
6000
4000
Time (ps)
8000
10000
Figure 3-9: Lattice surface temperature versus time predicted by the heat equation
and BTE with full relaxation times at T = 100 K. Ballistic effects are more apparent
at low temperatures where relaxation times are longer.
an earlier work by Koh and Cahill.9 4 In their work, a modulation-frequency dependence of the thermal conductivity of semiconductor alloys was observed, with a lower
thermal conductivity measured at higher modulation frequency. They explained this
result as follows: as the modulation frequency of the heating pump beam is increased,
the cross-plane thermal penetration depth Id becomes shorter, increasing the fraction
of phonons which have MFP longer than the penetration depth. The larger fraction
of ballistic phonons results in a smaller heat flux and hence lower effective thermal
conductivity, similar to that observed in our numerical simulations. The important
component of their interpretation is that the difference between the true and measured thermal conductivity is the contribution by phonons with A, > Lpd
=
2 x ld.
In other words, the effective thermal conductivity in the quasi-ballistic case is due to
phonons with A, < Lpd; phonons with A, > Lpd do not contribute to the thermal
conductivity measured by the experiment.
We can use this interpretation to better understand our numerical simulations.
As a first step, we examine the single-pulse BTE response; the accumulation effect of
multiple pulses will be considered later in the chapter. We fit the single-pulse BTE
solution with a Fourier's solution using an effective thermal conductivity keff and
interface conductance Geff, both of which will be smaller than the values specified in
the BTE computation due to quasi-ballistic effects. This fit is demonstrated in Fig. 310 for T=300 K. The BTE solution, in which k = 140 W/mK and G=1.1x108 W/m 2 K
were specified, matches a Fourier's law solution with different effective values of kef f
100 W/mK and Ge! f = 1.0 x 108 W/m 2 K. Under Koh and Cahill's interpretation, the
discrepancy between these two thermal conductivity values is due to phonon modes
with A, > Lpd. For the single pulse case, Lpd
2x
i/7ro, where a ~ 10-' W/m 2 K is
the thermal diffusivity of silicon at T=300 K and t ~ 5 ns is the approximate timescale
of the experiment. Using these values, we find that Lpd-
2.5pm, suggesting that
phonons with MFPs longer than about 3 pm do not contribute to the measured
thermal conductivity. To determine if this value is reasonable, we use the frequencydependent model of thermal conductivity described in section 3.3.4 to calculate the
thermal conductivity excluding phonons with MFPs longer than
3ptm. The result
0.041
0.0250.020.
E
0.015-0.010.0050
2
6
4
Time (ps)
8
10
X 04
Figure 3-10: Fourier law solutions for k=140 W/mK and k=100 W/mK compared to
the solution of the BTE for an aluminum film on silicon at T = 300 K. Even at room
temperature, the BTE solution does not match that predicted by diffusion theory.
Instead, the BTE solution almost exactly matches a Fourier's law curve with reduced
effective thermal conductivity, indicating quasi-ballistic transport is occurring.
is k ~ 100 W/mK, in remarkable agreement with the value obtained from fitting the
BTE solution.
The results of this fitting procedure for temperatures down to T=100 K are shown
in table 3.2. At and below T = 100 K a satisfactory fit using Fourier's law is difficult to
obtain for reasons that are presently unclear. However, above T=100 K, there is quite
good agreement between the MFP cutoff values Ac,, obtained from the frequencydependent model of thermal conductivity, and twice the penetration depth Lpd. These
results further support the hypothesis that phonons with A, > Lpd do not contribute
to the measured thermal conductivity.
We can now see that we have made good progress towards our goal of measuring
MFPs and relaxation times by studying the quasi-ballistic regime. The BTE calculation shows that when transport occurs over timescales comparable to relaxation
Table 3.2: Values of the effective (keff) and actual (k) thermal conductivities from the
BTE solution at several different temperatures, along with the MFP cutoff values A,
required to achieve this thermal conductivity and the effective penetration depth Lpd
at each temperature. The MFP cutoff values agree reasonably well with the effective
penetration depth.
T [K]
300
250
200
150
100*
* Poor fit of BTE solution
keff [W/mK]
100
130
170
230
450
k [W/mK]
140
190
273
429
815
A,
[pm]
2.5
3.0
4.0
5.3
20
L~d [pmrn]
2.4
2.8
3.6
5.1
8.7
times, the transport deviates from the predictions of heat diffusion theory. By analyzing the deviations with simple models, we are able to extract information about
phonon relaxation times. For the calculations in silicon, the results indicate that
phonons with MFP longer than approximately 3 pm contribute 40 W/mK to the
total thermal conductivity at room temperature. Unfortunately, if we consider only
the single-pulse response there does not seem to be any way to change the penetration
depth so that we can measure the MFP distribution at more than one point. This
issue will be solved in chapter 5, where we introduce an experimental technique which
can measure MFPs over a wide range of length scales and materials.
3.6
Unresolved puzzles
Despite the apparent consistency of our numerical analysis, there are two unresolved
puzzles. First, while the single-pulse numerical simulations suggest that quasi-ballistic
effects should be observable at room temperature in silicon at nanosecond timescales,
experimentally the correct thermal conductivity is routinely measured using standard
ultrafast techniques, though in these measurements the accumulation effect of multiple pulses is important. The reason for this discrepancy is presently unknown and
warrants further investigation.
Second, our simulation is presently not able to explain the modulation frequency
dependence of thermal conductivity observed in Koh and Cahill's experiment. So far,
we have only numerically solved for the single-pulse response of the BTE. In a typical
transient thermoreflectance experiment, however, an accumulation effect occurs where
the observed response of a sample is actually due to multiple pulses because of the fast
repetition rate (80 MHz, or one pulse every 13 ns) of the laser.9 5 Koh and Cahill's
observation of modulation frequency-dependent thermal conductivity can only be
interpreted using this multi-pulse response.
The multi-pulse response can be calculated from the single-pulse response using a
procedure described in Ref. [96]. Briefly, the procedure involves adding later portions
of the single-pulse response multiplied by a phase factor to the beginning of the
single-pulse response:
00
Z(t) =
h(qT + t)e0iwO(Tt)
(3.22)
q=O
where Z(t) is the multi-pulse response at time t, h is the single-pulse response, T is
the time between laser pulses, and wo is the modulation frequency. The modulation
frequency affects the multi-pulse response by changing the phase factor multiplying
the single-pulse response.
In the diffusive limit, the multi-pulse response should correspond to the same
thermal conductivity of the impulse response regardless of the modulation frequency
used. Therefore, if h is a Fourier's law response with a particular thermal conductivity,
then the multi-pulse response must correspond to the same value of thermal conductivity for all modulation frequencies. In order to measure a modulation-frequency
dependent thermal conductivity, then, the single-pulse response must deviate from
the Fourier's law response so that the multi-pulse response at different modulation
frequencies corresponds to different thermal conductivities. However, as can be seen
in Fig. 3-10, the BTE solution can be fit using a Fourier's law response with an effective thermal conductivity. Because the BTE response and Fourier's law response
with effective properties are essentially identical, the multi-pulse response of the BTE
must yield a thermal conductivity equal to its original effective value regardless of
(a) Amplitude signal, wo =15 MHz
3000
Time(pa)
4000
5000
6000
(c) Amplitude signal, wo =6 MHz
(b) Phase signal, wo =15 MHz
Time(ps)
6000
(d) Phase signal, wo =6 MHz
Figure 3-11: Typical multi-pulse responses at different modulation frequencies calculated from the single-pulse response for the BTE and Fourier's law. Both fits correspond to effective thermal conductivity keff = 100 W/mK and interface conductance
G = 1 x 10' W/m 2 K and therefore do not not predict a modulation-frequency dependence of thermal conductivity.
the modulation frequency. Thus, the present BTE calculation does not predict a
modulation-frequency dependent thermal conductivity.
This lack of frequency dependence is demonstrated in Fig. 3-11. Here, we have
used Eq. 3.22 to calculate the multi-pulse response of the BTE solution and the
Fourier's law solution with the effective properties keff=100 W/mK and G = 1 x 108
W/m 2 K. As the figure shows, this same set of effective parameters can explain both
the amplitude and phase responses of the BTE at two different modulation frequencies. We have calculated the multi-pulse responses at several different frequencies in
the MHz range and obtained the same result. Calculating the multi-pulse response
at lower modulation frequencies is difficult because up to 1 ys of the BTE solution
must be computed, but based on the above discussion we do not expect to observe a
frequency dependence even at these lower frequencies.
It is interesting to note that experimentally, Koh and Cahill did not observe a
modulation frequency effect in pure silicon. The effect was only observed for semiconductor alloys. It would therefore be useful to repeat the BTE calculations with
relaxation times corresponding to those for alloys and perform the same type of
analysis. Perhaps in this case the BTE would predict the experimentally observed
modulation frequency effect.
3.7
Conclusion
In this chapter we took a first step towards determining how we can measure MFPs
and relaxation times in materials. We showed that these properties are difficult to
measure in the diffusive regime because the carriers have already relaxed. Instead, we
must study the quasi-ballistic regime, where transport occurs on scales comparable
to MFPs and relaxation times. To further investigate this regime, we numerically
solved the frequency-dependent Boltzmann transport equation including a frequencydependent interface condition. We showed that quasi-ballistic heat transfer will differ
from the predictions of heat diffusion theory, and that by analyzing the deviations
with simple models we can extract very useful information about MFPs and relaxation
times. In chapter 5 we introduce a new technique to experimentally measure MFPs
using observations of quasi-ballistic transport with our ultrafast pump-probe system.
First, in chapter 4 we briefly describe our experimental pump-probe system and
the modifications we have made in recent years to make these measurements more
efficiently.
Chapter 4
Modifications to the pump-probe
experiment
4.1
Background
As discussed in the last chapter, studying heat transfer in the quasi-ballistic regime
allows us to extract information about phonon MFPs and relaxation times. To experimentally study transport in this regime, length and time scales must be comparable
to MFPs and relaxation times. This implies we need nanometer to micron spatial
resolution and picosecond time resolution. These requirements, though strict, can be
mostly satisfied using ultrafast pump-probe techniques. This type of optical experiment offers sub-picosecond time resolution and micron spatial resolution, which is
sufficient to study the quasi-ballistic regime in materials which have phonons with
MFPs in the micron range.
In this chapter we describe our experimental pump-probe setup arid the modifications which have been made since the setup was first constructed. Our experiment is
based on a pulsed Ti:sapphire laser with a fundamental harmonic at 800 nm and an
80 MHZ repetition rate. A schematic of the system is shown in Fig. 4-1.96 Because
the principle of the system and many experimental details are extensively described
in Ref. [96], only a brief review will be presented here.
The principle of pump-probe is as follows. A laser pulse is split into two pulses, a
Electro-Optic
Modulator (EOM)
Horizoital
8180
Crystal
ted filter
-
Vertica
Comipress
Sample
-
CCD
Camera
Polarizing
E-
r
4X
Expander
Cmr
Blue filter Waveplate
iglighl
Isltr------------I
loX
Objective
Cold Mirror
B.S.
300 mm
PIN
TI:Sapphire
Detector
Figure 4-1: Schematic of the pump-probe system in the Rosenhow Kendall Heat
Transfer Laboratory (from Ref. [96]). A Ti:Sapphire laser emits a pulse train of
800 nm, 250 fs pulses at a rate of 80 MHz. The pulse train is split into pump and
probe beams using a 1/2 waveplate and beamsplitter. The probe is directed to a
mechanical delay line and subsequently to the sample. The pump is modulated using
an electro-optic modulator (EOM), frequency doubled to 400 nm using a Bismuth
T riborate crystal (BIBO), and directed to the sample at the lower left of the figure.
The reflected probe light is send to a fast PIN detector (amplified or unamplified
photodiode); this signal is then sent to a lock-in amplifier.
pump and a probe. The pump pulse train is modulated at some frequency to allow
lock-in detection. In our implementation of the experiment, the pump pulses are
frequency doubled using a BIBO (Bismuth Triborate, Newlight Photonics) crystal to
allow easier removal of the pump pulses at the detector using color filters. A sample
consists of a substrate of interest coated with a thin
(~
100 nm) metal transducer
film, typically Al. The pump beam is focused to a spot of 1/e 2 diameter between 1060 pm, heating up the sample. This causes a change in the optical reflectivity of the
film, which is monitored using a variably time-delayed probe pulse train. The time
delay is accomplished by using a mechanical stage to create a variable difference in
path length. The probe has a 1/e 2 diameter of around 10 pm. For small temperature
rises induced by the pump beam, the change in optical properties of the film will be
linearly related to the change in temperature of the film. Therefore, by monitoring
the change in optical properties for different time delays of the probe pulses, we can
measure the temperature decay curve due to the impulsive heating from the pump
pulse. The shape of the temperature decay curve contains information about the
thermal properties of the system, so by fitting this curve to a thermal model, we
can extract the thermal properties of the material system. Typically, the interface
conductance G between the metal film and substrate and the thermal conductivity k
of the substrate are extracted.
Further details of the experiment and the thermal model are extensively discussed
in Ref. [96] and will not be presented here. Instead, we discuss the changes that
were made to make the system more stable and robust and simplify the task of data
collection.
4.2
Modifications to signal detection circuitry
One of the most helpful modifications we have made is improving the circuitry used
to extract the desired signal from the photodiode and send it to the lock-in amplifier.
To review, the pump pulse train is modulated at some frequency, typically between
1-15 MHz, to enable lock-in detection. This causes a change in temperature at the
surface of the metal film at the modulation frequency, which is detected by the probe
pulse train. Therefore, when the probe pulses are focused into the photodiode, the
resulting photocurrent contains very strong frequency components at multiples of 80
MHz, the repetition rate of the laser, and a much weaker component at the modulation
frequency which contains the information we wish to extract. This is very similar to
amplitude modulation of radio waves, with a lower frequency signal modulating an 80
MHz "carrier" wave. The challenge is to convert the photocurrent at the modulation
frequency, which may be as small as 1 pA, to a usable voltage which can be accurately
detected by the lock-in amplifier while removing the carrier signal.
In the first generation of the system, this challenge was made even more difficult
because of the way in which the modulation was performed. Originally the pump
beam was modulated with a square wave, which contains frequency components at
odd harmonics (3, 5, 7, ...) of the fundamental. Unfortunately, our lock-in amplifier
also mixes the incoming signal with a square wave, allowing these odd harmonics to
contaminate the desired signal at the fundamental modulation frequency. To remove
these odd harmonics, it was necessary to use a resonant LC filter with a high quality
factor. While this setup attenuated the odd harmonics and magnified the voltage
measured by the lock-in amplifier with a gain of 10, it also required the filter to be
changed in order to change the modulation frequency. This made data collection
laborious and time-consuming.
We solved this problem and removed the need for the resonant filters, allowing
us to change modulation frequencies completely automatically, as follows. First, we
replaced the original digital electro-optic modulator (EOM) amplifier (Conoptics 25D)
with an analog amplifier (Conoptics 25A). The analog amplifier amplifies whatever
waveform is sent to it, in contrast to the digital amplifier, which is capable of only
square wave (on-off) modulation. By modulating the pump beam with a sine wave
rather than square wave, we remove the odd harmonics, and hence the need for a
resonant filter.
Second, we use an active transimpedance amplifier (TIA) to convert and amplify
the photocurrent into a usable voltage signal.
Amplified detectors can be easily
purchased and provide a gain of ~ 2000 V/A, amplifying the small photocurrent into
the pV range. Our detector has a bandwidth of approximately 15 MHz, ensuring that
the signal at the modulation frequency is amplified while the 80 MHz carrier signal
is not.
These two modifications remove the need for the resonant filter and make changing
the modulation frequency completely automated. To make the system more robust
and stable, we make a few additional modifications. First, we place a band-pass filter
between the output of the amplified photodetector and the lock-in. This band pass
filter passes 20 kHz <
f
< 30 MHz, the typical range of the modulation frequency,
while further attenuating the carrier signal at 80 MHz. The removal of the interfering
carrier signal allows us to place the lock-in amplifier at a more sensitive setting,
making the whole setup less susceptible to external EM interference and internal
coherent pickup inside the instrument.9 7
Second, we moved the lock-in amplifier and signal generator as close to the detector
as possible to minimize the length of the coaxial cables. While coaxial cable ideally
should be shielded, a small amount of radiation at the modulation frequency can
still be radiated and picked up by signal cables.
This is particularly true for the
coaxial cables from the EOM, which transmit high voltages and thus radiate strongly.
Shortening the signal cables minimizes external EM interference and makes the system
far more stable. A picture of our cable setup is shown in Fig. 4-2.
Third, we added an automated beam blocker (DCH50 chopper, Electro-optical
Products Corporation, with USB-6008 DAQ board, National Instruments) to null
any constant offset voltages at each frequency. There are many sources of signal at
the modulation frequency which add a constant offset to the signal measured by the
lock-in amplifier, including radiation from EOM cables and internal pickup in the
lock-in itself. These offset voltages must be nulled, with the pump beam blocked,96
in order to obtain a measurement which can be directly compared to our model's
predictions. By automating the blocking of the pump beam, this cancellation can be
performed automatically, significantly easing the task of data collection.
Figure 4-2: Demonstration of good cable practices in our experiment. Note the short
cables from the photodiode and function generator to the lock-in amplifier.
4.3
Correction of laser beam astigmatism
The ideal fundamental mode from an oscillator cavity is a symmetric Gaussian beam.
A Gaussian beam is completely characterized by the beam diameter and the radius of
curvature.98 The diameter of the beam is typically characterized by the 1/e 2 diameter,
or the width of the beam where the intensity I = |E 2 has dropped to 1/e 2 of its peak
value, where E is the electric field. The radius of curvature specifies how fast the
wavefront is converging or diverging.
However, real lasers often exhibit astigmatism, meaning the beam diameter and
radius of curvature are different in different directions. In our laser, the beam exiting
the oscillator has diameters of 1.7 and 1.0 mm and beam divergence angles (related to
the radius of curvature) of 0.2 and 0.7 mrad in the X and Y directions, respectively.
This astigmatism makes tightly focusing the beam difficult. While the size of
the beam does affect the beam waist, the more difficult problem is the difference in
beam divergence angle. Specifically, this difference causes the Y beam waist to occur
approximately 75 pm behind the X beam waist, meaning the X and Y foci do not
Figure 4-3: Cylindrical lenses (focal lengths -80 and 130 mm) used to correct the
probe beam astigmatism.
99
occur at the same location. Thus, while the beam waists in each direction may be, for
example, 8 pm, the actual minimum waist that is achieved at any point is somewhat
bigger, and the beam is actually only round at exactly one point. In practice, this
point is very difficult to find.
Because the probe weights the metal film optical properties according to its shape,
the signal carried by the probe can depend strongly on its shape. This is particularly
true in materials with high thermal conductivity, where radial conduction is important
and the heat transfer induced by the pump and sampled by the probe will be affected
by the intensity profiles of each beam.96
Because our model assumes symmetric
Gaussian beams, it is important to make sure the experiment is consistent with the
model.
While in principle it should be possible to directly correct the output of the oscillator at the beginning of the beam line, we were not able to align the lenses correctly
to accomplish this. Since the pump beam is relatively circular due to other optics
along its path, we decided to correct the probe after the delay line to ease the alignment. We use a pair of cylindrical lenses to control the beam divergence angle in
one dimension, and hence the location of the beam waist in that dimension, so that
it more closely matches the beam waist location of the other dimension. Figure 4-3
shows the lenses used. These lenses slightly decrease the beam divergence angle so
that the Y beam waist occurs closer to the X beam waist. This circularizes the probe
beam to a diameter of approximately 11 pm.
One final point is that the BIBO crystal used to frequency double the pump beam
can severely distort the pump beam shape and prevent diffraction-limited focusing
if it is damaged. This was found to be the case for our 2 mm thick BIBO crystal.
We replaced this crystal with a 1 mm BIBO crystal to minimize the temporal and
spatial distortion of the pump pulse and were able to get diffraction-limited focusing
afterwards. The damage threshold for our BIBO crystal is 4 GW/cm 2 for a 250 fs
pulse at 800 nm. Care should be taken to ensure that the intensity remains well below
this value.
100
Sample preparation and film quality
4.4
One of the most important procedures to get reproducible, stable data from the
experiment is properly preparing the substrate and depositing a high quality metal
film. The primary requirement of the film is that the change in reflectivity be linearly
related to the change in temperature. This can only be satisfied if the incident pump
and probe intensities are not too high and if the film is tightly bonded to the substrate.
If any delamination occurs, or if the laser pulses damage the film, the data obtained is
essentially meaningless. We adopted the following procedure to ensure reproducible
results:
1. The sample surface must be cleaned thorougly to remove dust and organic
contaminants from the surface. For Si, an appropriate cleaning solution would
be a piranha solution (1:3 H20 2 :H2 SO 4 ). Of course, the solution will depend on
the particular substrate. If interface studies are desired, or if a high interface
conductance is required, it is especially important to remove the native oxide
layer prior to deposition. For Si, an 20:1 H2 0:HF dip is sufficient; this short
procedure increases the interface conductance by a factor of 3, from a 1 x 108
to a 3 x 108 W/m 2 K.
2. The film should be deposited under very high vacuum (< 10-6 torr) to minimize
impurities in the film. We have also found the electron beam evaporation consistently gives better films than does sputtering. The EBeamFP tool, a rapid
loadlock evaporator at MIT, is ideal for our purposes and can finish an entire
deposition procedure in under an hour. If film adhesion is a problem, a thin
(= 5 nm) Ti or Cr adhesion layer significantly improves the adhesion.
3. The power density of the pump and probe beams must not be too high or
else the film can be damaged, or the change in reflectivity will not be related
to the change in temperature. The film seems especially sensitive to damage
while under high vacuum. After much trial and error we have found the power
density limits are around 100mW/60pm 2 ~ 35 pW/pm2 for the pump and
101
20mW/10pLm
2
~ 60
[LW/pm 2
for the probe. If the pump power is too high, the
signal can be unstable and will not be constant in time. If the probe power is
too high, the change in reflectivity will not be linearly related to the change inl
temperature and the resulting data will not agree with the model. In practice,
we find that 10 pm is the minimum diameter for the probe beam which permits
enough power for a usable signal. Note that these values are approximate
because they depend strongly on the quality of the film.
4.5
Other miscellaneous modifications
In the new Rosenhow Kendall lab it was found that there were fairly strong air
currents blowing over the table from the ventilation system. These air currents cause
the pump and probe beams to wobble slightly with respect to each other, resulting in
increased noise. To remove this noise source, we surrounded the entire setup with a
clear plastic curtain. This makes the air much more still over the table, and has the
added benefit of reducing dust accumulation on the optical table. The present setup
is shown in Fig. 4-4.
We also found reflections from various optical elements, particularly from the
doubling crystal, could sometimes be reflected back into the laser and destabilize the
cavity. To prevent this, we placed the optical isolator directly in front of the output
of the oscillator. This significantly stabilized the laser and reduced laser intensity
noise.
4.6
Conclusion
In this chapter we have described our pump-probe experiment and the recent modifications which have made it more robust, stable, and automated. We also showed that
the system possesses the time and spatial resolution necessary to measure thermal
properties at scales comparable to phonon MFPs. In the next chapter, we introduce
a novel technique to experimentally measure MFPs using our pump-probe setup.
102
Figure 4-4: Picture of the curtains used to enclose the experiment and reduce air
currents and dust accumulation.
103
104
Chapter 5
Measuring mean free paths using
thermal conductivity spectroscopy
5.1
Introduction
In the previous two chapters we showed that studying transport at the length and
time scales of phonons can reveal important information about the underlying phonon
distribution, and that ultrafast pump-probe techniques are capable of the length and
time resolution required to observe these effects. In this chapter, we introduce a novel
thermal conductivity spectroscopy technique which is able to measure phonon MFPs
as a distribution like that plotted in Fig. 3-1 using our pump-probe experiment.
As mentioned in earlier chapters, despite the crucial importance of the knowledge
of phonon MFPs to understanding and engineering size effects, MFPs are largely
unknown for even bulk materials and few experimental techniques exist to measure
them. Traditionally, empirical expressions and simple relaxation time models have
been the only means to estimate MFPs."
Recent first-principles calculations in mate-
rials such as silicon show that MFPs of phonons relevant to thermal conductivity vary
by more than five orders of magnitude over the Brillouin zone. 86 '9 1 Experimentally,
inelastic neutron scattering has been used to measure phonon lifetimes in certain materials, but its resolution is limited and.it is not readily accessible. 9 9 A time-resolved
x-ray diffraction and time-domain thermoreflectance technique can measure ballistic
105
transport in some structures, and was used to study transport in GaAs.1 00 Koh et al.
observed a modulation frequency dependence of the thermal conductivity in a transient thermoreflectance experiment, which can lead to some knowledge of the phonon
MFP distribution, but this technique is limited by the modulation frequency. 94
In this chapter, we introduce a thermal conductivity spectroscopy technique which
can measure MFP distributions over a wide range of length scales and materials using
observations of quasi-ballistic heat transfer. The technique is based on the prediction
that the heat flux from a heat source will be lower than that predicted by Fourier's law
when some phonon MFPs are longer than the heater dimensions due to nonlocal heat
conduction external to the heat source.8 Our numerical solution of the Boltzmann
transport equation from Chapter 3 also showed this effect; the prediction was recently
confirmed experimentally in a transient grating experiment which measured the heat
transfer from nickel nanolines on a sapphire substrate using a soft x-ray probe. 93 The
authors observed higher thermal resistance than predicted by Fourier's law between
the nickel and sapphire substrate and attributed this observation to an additional
"ballistic resistance" due to ballistic transport external to the nickel nanolines. The
total thermal resistance was composed of contributions from both the nickel-sapphire
interface and the sapphire substrate. While this experiment measures the total thermal resistance, yielding an average MFP, our technique can distinguish the separate
interfacial and substrate resistances, enabling us to extract the thermal conductivity
contributions from phonons with different MFPs in the substrate. As mentioned earlier, considering the frequency dependence of the MFPs is essential because MFPs
vary over such a broad length range. In addition, our approach is based on a standard
pump-probe thermoreflectance technique, simplifying the experiment and modeling.
To see how the thermal conductivity spectroscopy technique works, let us examine
the transport of phonons in a transient thermoreflectance (TTR) experiment.
A
sample usually consists of a metal film on top of a substrate which we assume is a
dielectric. At t = 0 the metal film is impulsively heated by an ultrafast laser pulse
of 1/e2 diameter D, called the pump pulse. Phonons emitted by electrons diffuse
through the film and travel to the interface, where they are transmitted or reflected;
106
Figure 5-1: Illustration of the change from the diffusive transport regime (left), where
the heater size D is much larger than MFPs, to the ballistic regime (right), where
the heater size is much smaller than MFPs and a local thermal equilibrium does
not exist. The thermal conductivity spectroscopy technique measures the change
in thermal resistance as a function of heater size to determine the contributions of
different phonon MFPs to the thermal conductivity.
the transmitted phonons then travel through the substrate.
We are concerned with heat transport in the substrate.
As the transmitted
phonons travel through the substrate, their probability of scattering depends on the
value of the phonon MFPs A, relative to D, where w is the phonon frequency. If
D
>
A., the transmitted phonons scatter sufficiently to relax to a local thermal
equilibrium, allowing the use of Fourier's law. Alternatively, if D < As, phonons will
not scatter near the heated region and the transport will be ballistic. It has been
shown that in this regime, the actual heat flux is lower than the Fourier's law prediction, which assumes the occurrence of scattering events which are now not taking
place.8 93
' This reduction in heat flux is the origin of the ballistic resistance which
is observed experimentally.93 In real materials, the strong frequency dependence of
phonon MFPs A, means that for any value of D comparable to MFPs, the transport
will be quasi-ballistic: some phonons will be in the diffusive regime, others in the ballistic regime, and intermediate MFP phonons somewhere in between. The magnitude
of the ballistic thermal resistance will depend on the particular value of D relative to
the MFPs.
The above discussion shows that a measurement of heat transfer in the quasiballistic regime contains information about phonon MFPs. To see why, let us start
with the case D > A, and decrease D. For the case of large D, the heat transfer
107
is diffusive and described by Fourier's law. As D decreases, the ballistic thermal
resistance will appear when some phonons have A, > D, with the magnitude of the
ballistic resistance depending on the importance of a particular group of phonons
to the heat transfer. For example, if the ballistic resistance grows significantly as
D decreases from 60 pm to 30 pm, we can conclude that phonons with MFPs 30
pum < A, < 60 pm carry a significant portion of the heat. Our thermal conductivity
spectroscopy technique consists of systematically varying D and observing the change
in ballistic resistance from one value of D to the next, as illustrated in Fig. 5-1. By
doing so, we can infer the contribution of different phonon MFPs to the thermal
conductivity based on the change in ballistic resistance. The technique is similar to
point contact spectroscopy for electrons in which electron transport becomes ballistic
when the contact diameter is comparable to the electron MFP. 101
5.2
Experiment
We experimentally demonstrate this technique using transient thermoreflectance (TTR)
because TTR routinely achieves micron spatial resolution and picosecond time resolution, comparable to phonon MFPs and relaxation times in many materials. We
study c-Si at T < 300 K, because at these temperatures many phonon MFPs are
hundreds of microns or larger, making the pump beam waist D between 10 - 60 Pm
much smaller than a significant fraction of the phonon MFPs.
We performed a standard TTR experiment on a high-purity (> 20, 000 Qm) natural Si wafer coated with 100 nm of Al using electron-beam evaporation. The wafer
was cleaned using a piranha solution and the native oxide removed using an HF etch
immediately prior to deposition, giving a high interface conductance G = 3.5 x 10'
W/m 2K at T = 300 K. The thickness of the film was verified using profilometry. Our
experimental setup has been described elsewhere. 95 To obtain quantitative information from TTR it is necessary to fit the experimental data using a thermal model.
We use a two-dimensional model based on the heat equation,9 5 from which we can
obtain the interface conductance G between Al and Si and the thermal conductivity
108
a)
0.3
*
0.25
(-20)
b)
,
-15-
(pm D= 15
a25
S
0.2
D=15
gm w
-30
-35-
0.15 --
-
-4
6
4
D=_60pm
0.1 -2000
45 2500
3000
3500 4000 4500
Delay (pa)
5000
1000
5500
2000
3000
4000
Delay (ps)
5000
Figure 5-2: Experimental data (symbols) at T = 90 K for pump beam diameters
D = 60 pmn and D = 15 pm, along with the best fit to the a) amplitude and b)
phase data from a thermal model (solid lines). To show the sensitivity of the fit, the
model prediction for the best fit thermal conductivity values plus and minus 10% are
shown as dashed lines. The thermal conductivities obtained from the fits are 630 and
480 W/mK, respectively, different from each other and both far from the accepted
thermal conductivity of 1000 W/mK.'o2
k of Si. When the heat transfer is diffusive, this fitting procedure returns accurate
thermal properties for a variety of materials. 95 However, for quasi-ballistic transport
the experiment will measure an additional ballistic resistance, the magnitude of which
depends on the number of modes with MFPs longer than the pump beam diameter.
Figures 5-2(a) and 5-2(b) show representative experimental amplitude and phase
signals (R 2 = X 2 + Y 2; <P = tan-- (Y/X); where X and Y are the in-phase and
out-of-phase signals returned from the lock-in amplifier, respectively), and the fitting
curves which are used to extract the thermal conductivity. 95 The data in this figure
were taken at T=90 K for pump beam diameters D = 60 pm and D = 15 pm.
The fits are quite good, but correspond to thermal conductivities of 630 W/mK and
480 W/mK, respectively, different from each other and both far from the accepted
thermal conductivity of 1000 W/mK at 90 K. 1 02 This indicates that the heat transfer
in the substrate is significantly smaller than that predicted by Fourier's law.
Figure 5-3(a) shows our TTR measurements of the thermal conductivity of Si
10 2
and
versus temperature, along with literature values of the thermal conductivity
modeling results which will be explained later in the chapter. The symbols and error bars represent the average and standard deviation, respectively, of measurements
109
4
~a)
2
3.5 -
~b)
3-
103
Z 2.5
0 2
- 10
--- Calculated, A< 60 im
--- Calculated, A < 30 pm
0 1.5
--- Calculated, A-05pm
- Literature
STTR, D=60pm
2 TTR,D=30pm
A TTR,D=15gm
8
102
(K)
Temperature
0.5
.
a
0
50
TTR,D=60pjm
TTR, D=30Rm
A TTR, D=15 m
200
150
100
Temperature (K)
250
300
C)
1.2
1.1
0.9-
0
0
z
.8A
T=150K;D=60 m
T=60K;D=0±m
T=80 K; D=30 gm
> T=40K;D=15im
03
4
5
6
7
8
9
10
Modulation frequency (MHz)
11
12
Figure 5-3: (a) Experimental (symbols), calculated by only including the contributions of the indicated MFP or smaller (dashed lines), and the literature (solid line)
thermal conductivity of pure c-Si. 0 (b) Measured interface conductance of the Al/Si
interface. It is seen that the interface conductance does not depend on beam diameter. (c) Representative measurements of the normalized thermal conductivity versus
pump beam modulation frequency for several temperatures and beam diameters. No
frequency dependence of the thermal conductivity is observed.
110
taken at different locations on different samples, prepared at different times, at four
different modulation frequencies from 3-12 MHz. Three different pump 1/e
2
diame-
ters of 60 pm, 30 pm, and 15 pm were used with a constant probe beam diameter
of 11 pm. The ellipticities e of the pump and probe beams were 0.8 < e < 1. The
pump beam power was adjusted so that the intensity remained the same for each
diameter; the probe power remained constant. Care was taken to ensure that the
steady-state and transient temperature rise of the sample did not exceed 2 K. The
lateral spreading of heat in the Al film is estimated to be less than 1 Pm in the temperature range of the experiment. All measurements were taken under high vacuum
(1 x 10-5 torr). The temperature of the sample was monitored using a silicon diode
placed next to the sample. At room temperature, where MFPs are shorter, our measurements are independent of diameter and in good agreement with literature values,
but our measurements begin to diverge from the literature values below around 200
K and a diameter dependence of the thermal conductivity appears. This discrepancy
is because the ballistic thermal resistance measured by the experiment increases as
temperature decreases due to the rapid increase in MFPs. A smaller thermal conductivity is measured for smaller pump beam diameters because the ballistic resistance
is larger around a smaller heated region.
The thermal interface conductance values between Al and Si measured using different laser beam diameters are shown in Fig. 5-3(b), and no diameter dependence
is observed, indicating that ballistic transport in the silicon substrate is responsible
for the observed results. In the transient grating experiments in Ref. 93, the total
thermal resistance from the interface and substrate was measured, making it difficult
to study transport specifically in the substrate.
Koh et al. observed a laser modulation frequency dependence of the measured
thermal conductivity of semiconductor alloys in a pump-probe experiment and interpreted the result by assuming that some phonons are ballistic over the cross-plane
thermal diffusion length.94 We did not observe a strong frequency dependence as
shown in Fig. 5-3(c), suggesting that the change in radial heat transfer is the dominant effect rather than a one-dimensional thermal penetration effect. The authors of
111
Ref. 94 also did not observe a frequency dependence of the thermal conductivity in
Si at room temperature.
5.3
Discussion
To further understand our measurements, we need to examine how ballistic modes
affect the heat flux. In principle the transport in the quasi-ballistic regime can be
calculated using the Boltzmann transport equation (BTE)."1 However, solving the
BTE is difficult because the distribution function is a function of 5 variables under
the isotropic crystal approximation, and a large spatial domain comparable to D is
required. More importantly, the solution requires the MFPs, which are generally
unknown, to be specified. We would like to measure spectrally resolved properties
without any a priori assumptions concerning the relaxation times or MFPs.
Therefore, rather than solving the BTE, we make the following approximation
which still yields useful results. We divide the phonons in the substrate into two
groups, a diffusive group and a ballistic group. The diffusive group has short MFPs
A, < D so that a local thermal equilibrium exists among these modes and diffusion
theory is valid. The ballistic group, however, has MFPs A, > D which are assumed
to be effectively infinite. If we examine the BTE,81 we find that this assumption
completely decouples the ballistic group from the rest of the phonons; individual
phonon modes in the ballistic group propagate independently of the diffusive group
and of each other. As shown earlier, while these ballistic modes still carry heat,
due to their ballistic resistance, they carry significantly less heat than is predicted
by Fourier's law. Meanwhile, the diffusive group is transporting heat according to
Fourier's law. Under this approximation, then, the thermal conductivity measured in
the experiment is the thermal conductivity of the diffusive group with MFP A, < D;
the ballistic group does not contribute to thermal transport due to the effectively
infinite ballistic resistance. 94
To verify this model we can calculate the expected thermal conductivity if the
ballistic group, with A, > D, does not contribute to the measured thermal con112
ductivity. Because there are many different suggested relaxation times for Si in the
literature, we compute the phonon MFPs in Si from first principles, following a similar procedure as described in Ref. 86. Briefly, density functional theory is employed
to compute the potential energy derivative in Si crystals. Perturbation to the atomic
positions is used to extract the third and fourth order anharmonic coefficients of the
potentials, and Fermi's golden rule is used to compute relaxation times, while other
quantities such as the dispersion and group velocity are computed from the harmonic
force constants.io3 Using the computed information, the thermal conductivity can be
determined from k =
Cqv
,qrT,
where q is the phonon wavevector. The ballistic
group can be excluded by removing terms from the sum according to the MFP of the
mode. The results of the calculation are shown as the dashed lines in Fig. 5-3(a).
The agreement between our measurements and this simple cutoff model is reasonably
good. Obtaining quantitative agreement is difficult because at these low temperatures the thermal conductivity is highly sensitive to the isotope concentration, which
is not known exactly for our sample. We take a typical isotope concentration value
for natural silicon specified in chapter 3; the calculation using this value matches our
experimental results reasonably well.
We can now use our measurements to determine the thermal conductivity distribution of silicon as a function of MFP.10 4 In our experiment, we have measured the
thermal conductivity of the diffusive group of phonons with MFPs shorter than the
pump beam diameter. Thus, normalizing the measured thermal conductivities by
the true thermal conductivity of Si will give the fraction of the thermal conductivity
contributed by phonons with A, < D, which is simply the thermal conductivity accumulation distribution. Our experimental measurements of this distribution, along
with that predicted by the first-principles calculations, are shown in Fig. 5-4. The
agreement between the experiment and the calculation is again reasonably good. Note
that no assumption about the MFPs was used in the analysis of the experimental data:
the data were analyzed using the heat equation to determine the apparent thermal
conductivity, and this value was compared to the results of the first-principles calculations. The consistency between these two approaches is encouraging, and indicates
113
m
TTR, D=30 sm
A TTR, D=15 Rm
o 0.8- -T=60
K
o TTR, D=60
V
j
0.9 -
-
1
--
T=90 K
o00.7
. -T=150
K
E 0.6-
0.4 0.3-
E 0.1 0
10
10
10
MFP (gm)
10
10
Figure 5-4: Experimental measurements (symbols) and first-principles calculations
(lines) of the thermal conductivity accumulation distribution of silicon versus MFP.
The symbols and error bars represent the average and standard deviation, respectively, of measurements taken at different locations on different samples, prepared at
different times, at four different modulation frequencies from 3-12 MHz. Because of
the finite number of reciprocal space points included in the first-principles calculations, the calculated distribution of MFPs is discrete and cannot include some long
wavelength (and hence long MFP) modes. We use an extrapolation (dashed lines) to
estimate the contribution from these long wavelength modes.
114
that our technique is accurately measuring the thermal conductivity contributions
from different phonon MFPs.
5.4
Conclusion
In summary, we have demonstrated the first experimental technique which can measure the MFPs of phonons relevant to thermal conduction across a wide range of length
scales and materials. While empirical expressions and simple relaxation time models
have traditionally been the only means to estimate MFPs, our technique enables a
direct measurement of how heat is distributed among phonon modes. Although our
demonstration of the thermal conductivity spectroscopy technique is based on low
temperature experiments where a considerable fraction of the phonons have MFPs
longer than tens of microns, the technique can be extended to a much wider length
range by creating smaller heat sources.
There are many ways to achieve smaller
heat sources such as tight focusing of laser beams, transient grating experiments, and
lithographically patterned heaters or light absorbers. Considering the crucial importance of the knowledge of MFPs to understanding and engineering size effects, we
expect the technique to be useful for a variety of energy applications, particularly
for thermoelectrics, as well as for gaining a fundamental understanding of nanoscale
heat transport. For example, determining how much heat is carried by which MFPs
will allow the design of structures which can most effectively scatter these modes,
reducing the thermal conductivity of the material. For thermoelectrics, this would
directly translate to an increase in energy conversion efficiency, assuming the electrical
properties are minimally affected.
115
116
Chapter 6
Measuring mean free paths at the
nanoscale
6.1
Introduction
In the last chapter we introduced a thermal conductivity spectroscopy technique
which is able to measure phonon MFPs and their contribution to thermal conduction,
and we demonstrated the technique by measuring the MFP distribution of silicon at
cryogenic temperatures. The technique was successful in this demonstration because
phonon MFPs in silicon are hundreds of microns or longer at these low temperatures,
much longer than the pump beam diameters of tens of microns. The natural next step
is to measure MFPs using the technique in materials of technological relevance, like
thermoelectric materials at room temperature. Unfortunately, applying the technique
requires creating a heated region of size comparable to the MFPs of the heat-carrying
phonons, which are in the tens to hundreds of nanometers for many materials. This
small length scale is not accessible optically due to diffraction; the minimum beam
waist that can be typically achieved is approximately one micron. As discussed in
chapter 4, the practical minimum size for the pump and probe beams is much larger,
approximately 10 pm, due to restrictions on power density in the film. We therefore
have a problem if we wish to probe phonon transport at the nanoscale using this
technique.
117
Figure 6-1: Schematic of the metallic nanodot array on a substrate illuminated by a
much larger optical heating beam. Because the effective length scale for heat transfer is the dot diameter rather than the optical beam diameter, heat transfer at the
nanoscale can be observed, beating the diffraction limit.
In this chapter, we introduce a modification of thermal conductivity spectroscopy
which allows us to access these length scales. Rather than heating a continuous metal
film, in this modified technique we instead lithographically pattern metallic nanodot
arrays on the sample using electron beam lithography (EBL), as shown in Fig. 6-1.
Then, we use the pump beam to illuminate the entire dot array and the probe to
measure the heat transfer from the dots. In this way, the effective length scale for
heat transfer is the dot diameter rather than the optical beam diameters. Because
EBL is capable of spatial resolution down to tens of nanometers, we are able to probe
heat transfer at length scales far below what is possible optically. Here we study
phonon transport in sapphire as this material is transparent to visible light. A similar
technique was demonstrated in Ref. [93], where soft x-ray light was used to observe
quasi-ballistic transport from thin Ni lines patterned on a sapphire substrate. In
this work, we show we can observe quasi-ballistic transport simply by using transient
118
thermoreflectance (TTR), and use our observations to extract the MFP distribution.
This chapter is organized as follows. First, we introduce the model used to calculate the heat transfer from the dots. Next, we discuss the details of our experiment
and the fabrication of the dot arrays. We then present our measurements of the MFP
distribution, and discuss the possibilities for future work.
Heat Transfer Model
6.2
6.2.1
Introduction
The heat transfer analysis used for our experiment has been described in detail in
Ref.
[95]
and will be briefly reviewed here. The solution of the heat equation which
accounts for radial heat transfer, Gaussian beams, and the high repetition rate of our
laser is given in terms of the transfer function Z:95
00
Z(t) =
[
(6.1)
H (wo + mw) ej)s
m=-oo
The real and imaginary parts of Z correspond to the in-phase and out-of-phase signal
return by the lock-in amplifier, respectively. 95 Here wo is the angular modulation frequency, typically between 1 and 15 MHz; w, = 80MHz is the angular laser repetition
frequency;
j
=
1
, and H(w) is the solution of the heat equation.
The solution H(w) can be written as the product of the transfer matrices for the
layers.1 1 5 The properties at the bottom of a layer can be related to the properties at
the top of a layer using a transfer matrix given by:
cosh(qd)
Ob
f1
where 0 is the temperature,
;-gsinh(qd)
L
-o-zq sinh(qd) cosh(qd)
f
0,(62
f1
is the heat flux, subscript b and t denote the bottom
and top of the layer, respectively, d is the layer thickness, Uz the cross-plane thermal
conductivity, and q2 = jw/a, where a is the thermal diffusivity. Multiple layers can
119
be conveniently treated simply by multiplying the matrices together:
Ob
..
A B
C D
fAj
O
(6.3)
LAf
Mi is the transfer matrix for layer i. If the bottom layer is semi-infinite or adiabatic,
which is usually the case, then the surface temperature at the top of the layers, Ot, is
given in the terms of the heat flux boundary condition ft as:
Ot =
(6.4)
(ft
C
To account for radial conduction, a zero-order Hankel transform is applied to the heat
equation in cylindrical coordinates. It has been shown that the results above apply
in this case with only a modification to the definition of q, which now becomes: 95
q2
_rk
2
(6.5)
CvjW
Orz
where o, is the in-plane thermal conductivity, C, is the volumetric specific heat, and
k is the Hankel transform variable. The heat transfer boundary condition ft is now
a Gaussian function of the radial coordinate r, following the intensity distribution
of the pump beam of radius wo. Weighting this result with the Gaussian intensity
distribution of the probe with radius wi gives the final solution to the heat equation
H(w):95
H(w) = j
00
k
() (
(-D
exp
(-k
2 (W2~
w±
))
dk
(6.6)
Because the final solution is normalized to facilitate fitting with experimental data,
the constants have been omitted from the above equation. This is the model which
is used for a typical pump-probe experiment.95
6.2.2
Single Dot Heat Transfer Model
The previous analysis models the heat transfer through continuous layers with a
Gaussian heating and probing profile. This analysis must be modified to account for
120
the discontinuous nature of the dot array. We first assume that the heat transfer
from a particular dot is independent from the other dots, making the modification
is particularly simple. Since the dot radius a is much smaller than the pump and
probe radii, both the heating and probing profile can be well described by a radial
step function of radius a. The zero-order Hankel transform of a step function is
JI(ak)/k,106 where Ji(z) is a first-order Bessel function of the first kind, and so
replacing the exponentials in Eq. 6.6 with this function gives H(w) for a circular dot:
H(w) =
j
k
(
Jj2(ak)dk
(6.7)
If the dot were square rather than circular, we would obtain a similar mathematical
expression but with a two-dimensional integration over a sinc(x) x sinc(y) function.
6.2.3
Dot Array Heat Transfer Model
If the dots are sufficiently close together that the heat transfer from a particular dot is
affected by the presence of other dots, then it is necessary to solve for the heat transfer
from the entire dot array. Fortunately, the dot array can be incorporated into the
transfer matrix method fairly easily. To account for the discontinuous nature of the
dots, we model the metal film as a special layer in which the thermal conductivity in
the r direction, Ur, is zero. Then, we apply a heating profile that has the shape of
the dot array. In this way, the special film is only heated at the location of the dots,
and, because Or = 0, the heat can only diffuse in the z direction. Thus there will
be no heat transfer between the heated regions, corresponding precisely to the actual
physical system.
The next step is to determine the appropriate heating profile.
A square array
of circles is difficult to Fourier transform analytically, but a square array of squares,
or a two-dimensional square wave, can be represented exactly as a Fourier series.
Therefore, we model the heating profile as a two-dimensional square wave with each
square having a side length of w = 2a and a period L. It is now more appropriate to
solve the heat equation in Cartesian rather than cylindrical coordinates, and so the
121
definition of q becomes:
2
q =
OxY(k2+ k 2) +CjW
(6.8)
± k
where oxy is the in-plane thermal conductivity, and kx and ky are the spatial Fourier
transform variables in x and y, respectively.
To obtain H(w) we can perform the procedure as before; compute the transfer
matrices for different values of kx and k., compute the Fourier transform of the
heating function ft, obtain 6 t = (-D/C)ft, and weight this by the probe function to
get H(w). Since the square wave heating function is periodic, its Fourier transform
consists of discrete multiples of the fundamental spatial frequency Qo = 27r/L:
ft(Q) = Q
Sn Sm Xnm6(Q -
(6.9)
n~o)6(Q - mQo)
where Q is the energy of the pump beam, Q is a continuous spatial frequency variable,
and Xnm are the Fourier components of the square wave, given by:
n= m= 0
w2/IL2
-
Xnm=
iL(
1
- exp(-jmQo))
2 Tmj
Xnm ( 1 - exp(-jnQo))
-(1-exp(-jmQo)(1-exp(-jno)
47r 2nm
n =0,m
0
n # 0,m = 0
n
(6.10)
m5 0
Because ft only contains discrete multiples of Qo, Ot = -(D/C)ft only contains these
frequencies. It can be shown that weighting Ot by the same probe profile results in
the simple expression for H(w):
H(w) =
Xnml 2
n
(-)
(6.11)
n,m
m
where the subscripts n and m correspond to evaluating the argument at frequencies
kX = nQo and ky = mQo, respectively.
Note that this procedure has neglected the Gaussian intensity variation of the
pump and probe beams. To account for this, it is necessary to modify the heating and
probing profile by multiplying the square wave by a Gaussian. In the Fourier domain,
122
the multiplication becomes convolution, and the result is essentially a Dirac comb of
Gaussians at each discrete frequency point. The resulting Fourier transform is now
a continuous function of frequency because the Gaussian is aperiodic, significantly
complicating the analysis.
An approximation can be made if the squares are much smaller than the pump
and probe radii. Intuitively, if the pump and probe radii wo and wi are much larger
than the array period L then it is reasonable to approximate the pump and probe
radii as infinitely large and with uniform intensity. In the frequency domain, this can
be explained because of the mismatch in spatial frequencies. An array with a small
period L, relative to wo and wi, will have a high spatial freqency Qo = 27r/L. The
frequency components in the Gaussian are distributed as exp(-wQ 2 /8), where i=1
or 2 corresponds to the pump or probe radius. Thus if (wi/L)2 > 1, the frequency
components of the Gaussian will be confined to a region in frequency space close to
the discrete frequency of the square wave. In this case we can simply approximate the
function as a delta function and recover the simpler result given in Eq. 6.11. Because
the condition for this approximation to hold scales as (wi/L)2 , the approximation
is expected to be accurate even for periods approaching the probe radius wi. As
described in the next section, the probe radius is around 5 pm, and we are typically
concerned with L < 2 pum, and so the approximation is expected to be valid.
Finally, we note that in the previous two sections, we assumed a different shape
for each heat transfer model, a circle and a square, due to mathematical convenience
for each model. While it is possible that the shape of the nanodot could affect the
heat transfer, experimentally we find the heat transfer is a strong function only of
the dot size, not the shape.
6.3
Sample Design and Fabrication
The metal nanostructures are fabricated using a standard metal lift-off process. Before
discussing the fabrication details we quickly discuss the design of the dot arrays
which we found necessary to obtain usable data. Ideally, we would like to make
123
the dot diameters as small as possible so that we can study heat transfer at the
smallest length scales. This requires a dot thickness smaller than the dot diameter so
that the dot aspect ratio is not too large and the dots can adhere to the substrate.
Motivated by this goal, we initially tried fabricating dots with thicknesses around
20-30 nm but encountered several challenges. First, when the metal layer is so thin
the thermal conductivity extracted from the model is extremely sensitive to the metal
layer thickness. A difference of only 2-3 nanometers in the dot thickness can change
the fitted thermal conductivity value by up to 30 %. Furthermore, because electronbeam evaporation is an inherently random process, spatial variations in the deposition
rate and roughness in the film can easily lead to a difference in thickness between
dots on the order of a 2-3 nanometers. These two issues make obtaining reliable,
quantitative data from thin dots difficult.
Thin dots present another problem. The rate at which the dots cool is linearly
related to their volumetric specific heat. If we perform a simple lumped capacitance
analysis, we find the time constant for the temperature decay is
T
= Ct/G, where C
is the volumetric specific heat, t is the thickness of the dot, and G is the interface
conductance. If the dots are thin, their temperature, and hence the thermoreflectance
signal, quickly decays to a level that is at the noise floor of our experiment. Thus after
a certain delay time, we are only able to measure noise rather than the desired signal,
making it impossible to extract the thermal conductivity. Increasing the thickness to
70-80 nm increases the thermal time constant and makes the measurement possible.
From these considerations, we focused on obtaining reliable, quantitative results
from dots with thicknesses of approximately 70-80 nm and diameters larger than 100
nm. The sample fabrication procedure is as follows. The substrate is a 12.5 mm
square, single side polished, single crystalline c-plane sapphire wafer (MTI corporation). Sapphire is chosen because it is transparent to visible radiation and has MFPs
in the hundreds of nanometers.93 The sapphire wafer is rinsed with DI water, then
spin-coated at 1000 RPM with 950K PMMA resist (950K A2 from Microchem, A4
can also be used to give a thicker resist layer), giving a resist layer approximately
150 nm thick. The resist layer should be at least twice the desired metal thickness,
124
but not too thick as this decreases the resolution of the lithography process. Because
sapphire is electrically insulating, it is necessary to deposit a 5-10 nm Ti conduction
layer for the electron-beam lithography (EBL) process. The dot array is designed
using the CleWin4 layout software. The resist is exposed using EBL at 10 kV, permitting feature sizes down to 100 nm to be reliably written. Because the MIT EBL
machine often exposes a larger area than specified in the layout, a size matrix of
several slightly smaller diameter dots (typically 25-50 nm smaller than the desired
diameter) is exposed for each desired diameter. Next, the titanium layer is removed
using a 20:1:1 H 20:HF:H 20
2
solution, and then the resist is developed with a 3:1
IPA:MIBK solution. The wafer is immediately coated with a 5 nm Ti adhesion layer,
followed by 60-70 nm of Al, using electron-beam evaporation. Finally, the remaining
resist is removed by soaking the sample in Microstrip (N-Methyl-2-pyrrolidone) at 80
C for two hours, followed by a one minute low power sonication. The thickness of the
dots is verified using profilometry.
The final result is shown in optical and SEM images in Fig. 6-2. The nanostructures consist of arrays of closely packed circles or squares of varying diameter. Because
the dots are discontinuous and cannot exchange heat as they would in a metal film,
the effective length scale for heat transfer is the dot diameter rather than the optical
beam diameter. While ideally each dot cools independently of the others, when the
separation between the dots is comparable to the thermal penetration length lth the
heat transfer from a particular dot will be affected by presence of the other dots.
Because lh ~ fa/fo
-
1 im. in sapphire, where a is the thermal diffusivity and fo
is the modulation frequency, it is expected that for dot separations in the hundreds
of nanometers the dots will not be independent, necessitating the use of the model
derived in section 6.2.3.
When the dot radius becomes comparable to the dominant phonon MFPs in sapphire the transport becomes quasi-ballistic.
By observing the increase in ballistic
thermal resistance as the dot radius shrinks we can measure the contribution of different phonon MFPs to the thermal conductivity, just as we did to measure the silicon
MFP distribution in the previous chapter.
125
(a) Optical image at 1oX magnification.
(b) SEM image at 5 kV, 4000X magnification.
Figure 6-2: Optical image and SEM image of the nanodot array fabricated by electronbeam lithography (EBL). The numbers in the optical image correspond to the diameter of the dot (800 = 800 nm, and so on). Because the MIT EBL machine often
exposes a larger area than specified in the layout, a size matrix of several different diameter dots was exposed for each intended diameter; these different sizes are
indicated by the number of large dots below each array.
6.4
Results
We created both circular and square individual dots with diameters from 1-10 pm,
along with dot arrays containing square dots with diameters from 100 nm - 10 pm.
The measurements were taken at a modulation frequency of 12 MHz to shorten the
thermal penetration length and minimize the influence of other dots on a particular
dot's heat transfer. In order to obtain reliable data with a sufficient signal to noise
ratio (SNR), the pump and probe powers and sizes needed to be adjusted. Because
the dots occupy 25% of the surface area on the sapphire, only approximately 25% of
the reflected light is due to the dots and thus the signal is nominally four times smaller
than it would be with a continuous film. In practice, we find that the signal is often
up to ten times smaller than in the continuous film case because some probe light
is diffracted and does not reach the detector. To offset this decrease, we increased
the probe power, which increases the power reaching the detector but also decreases
the power in the pump beam. To offset this decrease in pump power, we shrink
the pump beam diameter to around 30 pm, increasing the pump beam intensity
126
1.2
'
~1
4)
0
\
0.8
O
0.6
-
S
0.8'
0
200
400
4000
5000
E
<0.4
0.2
0'
0
'
1000
2000
3000
Time (ps)
6000
Figure 6-3: Typical amplitude experimental data for D=400 nm Al dots on a sapphire
substrate. The inset shows data for shorter delay times, where oscillations in the signal
are observed. The oscillations are believed to be due to a transient diffraction grating
effect caused by thermal expansion and contraction of the dots.
and therefore the magnitude of the desired signal.
These modifications result in
a sufficiently high SNR to obtain quantitative measurements of thermal properties.
Figure 6-3 shows a typical experimental data curve. The early part of the decay
curve, up to approximately 2 ns, shows an interesting oscillation. Examining the
period of the oscillation, we find that the period is too long to be caused by acoustic
9 6 Instead, it seems likely
strain waves traveling through the thickness of the dots.
that this oscillation is a type of transient grating effect due to thermal expansion and
contraction of the dots. The dot array, as a periodic array of objects with a spacing
on the order of the laser wavelength, acts as a diffraction grating for the pump and
probe beams. When the pump heats up the dots, the dots thermally expand and
127
0.7-
25
--
0.5
D=10 pm .
0.5
-35
0.4-
D=-10 4M
40
D=400 nm
0.3-
.
.
1000
2000
.
.
.-
3000
4000
5000
Delay(pa)
45
1000
2000
3000
4000
Delay(pS)
5000
Figure 6-4: Experimental data (symbols) dot diameters D = 10 pm and D = 400 nm,
along with the a) amplitude and b) phase fit to the data from a thermal model (solid
lines) and 10% bounds on the fitted thermal conductivity value (dashed lines). The
thermal conductivities obtained from the fits are 37 and 22 W/mK, respectively. The
value for the 10 pm dots is in good agreement with the literature value for the thermal
conductivity of sapphire, but the value for the 400 nm dots is lower, indicating that
MFPs are in the hundreds of nm in sapphire.
subsequently. contract in an oscillatory manner, resulting in a small change in the
amount of probe light diffracted and thus the amount of light reaching the detector.
We estimate that for a 10 K temperature rise the dots expand by only 1-5 Angstroms,
a very small change which remarkably appears to produce the oscillating signal shown
in Fig. 6-3. After the oscillation damps out, the monotonically decreasing signal is
due to the thermal decay of the dots. It is this portion of the signal which contains
information about the heat flux from the dots to the substrate and hence the thermal
properties of the substrate.
Figures 6-4a and 6-4b shows the data and the model fit for 10 pm and 400 nm
dots. Recall that the in-phase signal X = Re(Z), out-of-phase signal Y = Im(Z),
amplitude R 2 = X 2 + Y2, and phase <D = tan-1 (Y/X), where Z is the transfer
function. The model fits both data sets quite well, but the fit for the 400 nm dots
corresponds to a smaller thermal conductivity of 21 W/mK rather than the bulk value
of 35 W/mK, indicating the heat transfer is in the quasi-ballistic regime for the 400
nm dots.
The measured thermal conductivity for all the diameters is shown in Fig. 6-5.
At the largest length scales of 1-10 pm, the heat transport is diffusive and the dots
128
45
40-
2 30-
025-
E
-
20-
1510
101
100
Diameter (pm)
Figure 6-5: Measured thermal conductivity of sapphire versus dot diameter. The
measurements indicate that the dominant heat-carrying phonons in sapphire have
MFPs < 1 pm.
cool independently; both heat transfer models return the bulk thermal conductivity
of sapphire, which is approximately 35 W/mK along the c-axis at room temperature. We also confirm that the shape of the dot has little effect on the heat transfer.
Deviations from Fourier's law appear for dot diameters less than 1 pm, where we
measure a lower thermal conductivity than the literature value. This indicates that
the transport is quasi-ballistic and the experiment is measuring an additional ballistic
resistance, exactly as was observed in chapter 5. Our results show that the dominant
heat carrying phonons in sapphire have MFPs < 1 pm, which is consistent with previously reported results, 9 3 and that phonons with MFPs between approximately 400
nanometers and 1 pm contribute slightly under half to the total thermal conductivity.
129
6.5
Conclusion
In this chapter we have extended the technique of thermal conductivity spectroscopy
to the nanoscale. By using electron-beam lithography to pattern nanodot arrays, we
are able to observe quasi-ballistic heat transfer and measure the MFP distribution
at length scales far below the diffraction limit. In sapphire, we have confirmed that
the dominant phonon MFPs are in the hundreds of nanometers. We expect that
this combination of ultrafast optical techniques with nanoscale spatial patterning will
prove useful in measuring MFP distributions in technologically relevant materials like
thermoelectrics, where MFPs may be in the tens or hundreds of nanometers.
130
Chapter 7
Summary and future directions
7.1
Summary
Nanoscale heat transfer is a fascinating and rapidly growing field which could make
substantial contributions to addressing one of our present energy challenges. In particular, nanostructured thermoelectric materials could allow some of the enormous
amount of wasted heat to be recovered as electricity, improving the overall efficiency of
our energy conversion systems. Despite recent improvements in the figure of merit of
nanostructured thermoelectrics, much remains unknown concerning the precise mechanisms by which the heat carriers in thermoelectric materials, electrons and phonons,
are affected by nanostructures. This thesis has explored the physics of nanoscale
transport in these materials and helped identify how we can use nanoscale effects
to engineer more efficient thermoelectric devices. Using a combination of modeling
and experiment we were able to better understand electron and phonon transport in
nanostructured thermoelectrics, and we introduced a new experimental technique to
help us understand fundamental phonon transport in all types of materials.
Chapter 2 studied electron and phonon transport in nanocomposite thermoelectric
materials using a newly developed grain boundary scattering model and materials
characterization.
Using the model we are able to explain how the thermoelectric
properties are affected in nanocomposites as well as to identify strategies which could
lead to more efficient materials.
131
The rest of the thesis focused on understanding the quasi-ballistic phonon transport regime, where a lack of scattering results in the loss of local thermal equilibrium,
and determining what we can learn from studying this regime. Ultimately, we showed
that observations of quasi-ballistic transport can yield extremely useful information
about phonon MFPs and their contribution to thermal conduction. In Chapter 3 we
numerically investigated this regime using the frequency-dependent phonon Boltzmann transport equation, showing that studying how the transport deviates from
diffusion theory can yield information about phonon relaxation times and MFPs.
Chapter 4 briefly discussed the modifications that were made to the pump-probe
system used in this work to experimentally study quasi-ballistic transport.
Chapters 5 introduced a new thermal conductivity spectroscopy technique which
for the first time is able to measure MFPs relevant to thermal conduction across a
wide range of length scales and materials. We demonstrated that by systematically
changing the size of a focused beam at length scales comparable to MFPs and observing the change in thermal resistance, the thermal conductivity contributions from
phonons of different MFPs can be determined. We presented the first experimental
measurements of this distribution in silicon at cryogenic temperatures (30 K to 100
K).
In Chapter 6, we extended thermal conductivity spectroscopy to the nanoscale
by observing heat transfer from lithographically patterned metallic light absorbers
with diameters in the hundreds of nanometers range. Accessing this length range is
important because many technologically relevant materials like thermoelectrics have
MFPs in the deep submicron regime. We demonstrated this modified technique by
measuring phonon MFPs in sapphire.
Considering the crucial importance of the knowledge of MFPs to understanding
and engineering size effects, we expect the techniques developed in Chapters 5 and
6 to be useful for gaining a fundamental understanding of nanoscale heat transport,
and for applications involving nanoscale thermal engineering such as thermoelectric
materials and heat transfer in microelectronic devices.
132
7.2
Future work
One important conclusion of this thesis is that studying transport at the length
and time scales of heat carriers yields far more detailed and useful information than
can be obtained from a typical macroscopic measurement.
In a sense, then, the
work presented here is only the beginning because it describes the development and
demonstration of one experiment which studies phonons at the MFP length scale, our
thermal conductivity spectroscopy technique. With this particular technique, there
is no shortage of interesting materials, both bulk and nanostructured, to which the
technique can be applied; and there are many possibilities to study electrons, phonons,
and photons at these same small scales to reveal interesting and useful results.
The future work for thermal conductivity spectroscopy consists of measuring
MFPs for increasingly complex materials. The simplest next step would be to perform a similar analysis as was done for silicon to other materials which have suitably
long MFPs at low temperatures. Ideally, the material would be simple enough to
permit a comparison of first-principles calculations and experimental results. III-V
materials such as GaAs would be suitable for this purpose. Analyzing the differences
between the MFP distributions of different materials can yield information about the
scattering mechanisms such as phonon-phonon scattering in pure semiconductors.
Next, one can consider the effect of alloying or doping on the MFP distribution.
For alloys, it is generally accepted that high frequency modes are strongly scattered
and a larger fraction of the heat is conducted by low frequency, long MFP modes.
Our technique should be able to confirm this picture and give an idea which MFPs
are carrying the heat in alloys. This type of information would be very useful for
thermoelectrics as they are almost always alloys.
After this, nanostructured materials can be investigated. It is believed that grain
boundaries are effective in scattering long MFP phonons, but exactly which phonons
are still carrying the heat in nanocomposite materials is not known. This type of
investigation could give results which are very useful for thermoelectrics. By determining which phonon modes carry how much heat, we will be better able to determine
133
how to effectively scatter these modes and reduce the thermal conductivity. As discussed in chapter 1, a reduction in thermal conductivity will lead to an increase in
energy conversion efficiency, assuming a minimal reduction in electrical properties.
At this point, when we start investigating materials with low thermal conductiv-
ities and short MFPs, an additional modification of the technique will be necessary.
As discussed in Chapter 6, accessing short length scales optically requires using lithographically patterned nanodot absorbers. We used this technique to measure submicron MFPs in sapphire. An important detail of this technique is that these nanodot
absorber arrays typically cover at most 25% of the sample surface area, assuming
a 50% duty cycle in the X and Y directions. The reason this technique worked for
sapphire is that sapphire is transparent to visible radiation, meaning the 75% of the
pump and probe light which reached the sapphire was not absorbed and thus did not
contribute to the measured signal. However, most materials of interest are opaque to
visible radiation. If we apply this technique to silicon, for example, 75% of the signal
will be due to silicon and only 25% will be due to the nanodot arrays. Since we do
not know a priori what the signal from silicon will be, it is difficult to extract the
desired dot signal from the total signal.
To overcome this problem, we have been investigating designing special nanodot
arrays which are suitable for studying heat transfer but also have special optical
properties which prevent the laser radiation from being absorbed by the substrate.
One idea is to design a structure which has has a very high reflectance for a particular
wavelength and polarization so that the incoming light is reflected before it can be
absorbed by the substrate. Determining how we can achieve strongly confined heating
despite the presence of a possibly absorbing substrate is an important hurdle which
must be overcome.
For electrons, a technique called point contact spectroscopy,101 similar to what
has been developed here for phonons, may yield information about electron MFPs.
This technique measures electrical conduction from nanometer-sized contacts which
are shorter than electron MFPs. By analyzing the deviations in the resistance from
the Ohm's law prediction, information about MFPs can be extracted. This technique
134
may be helpful in determining more directly than the models presented in Chapter 2
how electrons are affected by grain boundaries.
Apart from investigations of electron and phonon transport for thermoelectrics
applications, an important result of this thesis is that the pump-probe technique is
versatile and can be applied to many different experimental configurations. Traditionally, pump-probe for heat transfer applications was employed only to measure
thermal conductivities and interface conductances of different bulk materials using
the familiar metal film on substrate geometry; in chapter 6, we modified this geometry to measure heat transfer at the nanoscale using dot arrays. In this spirit,
there are myriad possibilities for how the same experiment can be applied to yield
novel and useful results. Some of the possibilities are using pump-probe to measure
heat transport down ultra-drawn polymer chains, measuring boiling and condensation
heat transfer at ultrafast timescales, and using AFM techniques with pump-probe to
achieve true nanoscale spatial and ultrafast time resolution.
These are just a few of the possibilities which could yield important and useful
information about nanoscale transport. As the length scales of technology continue to
shrink, our ability to understand and engineer transport at these scales will become
the enabling factor for future innovation. We hope that this thesis has given insight
into thermal transport at the nanoscale and will be useful for engineering the next
generation of nanostructured materials.
135
136
Appendix A
Numerical constants used in the
nanocomposite model
Values of the parameters used for the calculation of the electronic and thermal properties using the Boltzmann transport equation in Chap. 2 are given here. All data are
from Ref. [60] unless otherwise noted. While most values are taken directly from the
literature, four parameters require some modification. The low frequency dielectric
constant co, needs to be increased from its literature value to reduce the strength
of ionized impurity scattering. This is a common adjustment when ionized impurity scattering is a dominant scattering mechanism, and Vining5 1 also performed this
adjustment.
The second adjusted parameter is the non-parabolicity a of the X conduction
band. This parameter is not known with certainty and is often adjusted to obtain a
better fit. In this case, a value of az = 0.5 eV- 1 has been reported,1 0 7 but we find
that a higher value of 1.25 eV- 1 is necessary to explain the data.
The final adjusted parameters are the valence band deformation potential D, and
hole effective mass m*. For the deformation potential, here is no clear consensus on
the precise value, and reported values vary from 2.94 eV in Ref. [51] to 5 eV in Ref.
[49]. We find that a value in the middle of these, 4.0 eV, is required to fit the data
for our model.
The hole effective mass is difficult to define due to the warped shape of the heavy
137
hole band and perturbations from the split-off band. We find a value of m* = 1.2me is
necessary to fit the data, similar to the value of 1.4 me used by Vining5 1 and consistent
with Slack and Hussain's discussion.5 2 As discussed in Sec. 2.5, a higher value of 1.55
me is necessary to explain the nanocomposite data.
The temperature dependence of the band gap Eg(T) was accounted for using a
curve fit from Ref. [108]. The fit is given as follows:
Eg,si =
Eg,Ge
Eg(T)
=
1.1695 - 4.73 x 10-4T 2 /(T + 636)
0.85 - 4.774
x
10- 4T 2 /(T + 235)
= E,si(1 - x) + Eg,Ge
Here T is in K and Eg is in eV.
138
-- 0.4x(1 - x)
(A.1)
Table A.1: Lattice parameters for Sii_,Ge, used to calculate the lattice thermal
conductivity.
Comment
Value
Symbol (Units)
Property
a (A)
5.431(1 - x) + 5.658x
Lattice constant
Ci (1010 N/m 2 )
9.8 - 2.3x
Bulk modulus
640
- 266x
(K)
ED
Debye temperature
3
2329 + 3493x - 499x 2
p (kg/m )
Density
Ref. [65]
2.0
Ratio of normal to
3
umklapp scattering
Ref. [65]
1.0
(nAnharmonicity
type)
0.7
Anharmonicity
(ptype)
Ref. [65]
100
Strain parameter
1.4
Higher-order phonon
scattering exponent
139
Table A.2: Band structure parameters for Sii_,Ge, used to calculate electrical properties.
Comment
Value
Symbol (Units)
Property
Electron
longitu-
dinal/transverse
effective mass (X)
longituElectron
dinal/transverse
effective mass (L)
Energy gap between X
and L
Hole DOS effective
mass (bulk/nano)
Non-parabolicity of X
Low frequency dielec-
m*/m* (e)0
m*/m* (me)
1.59/0.082
EL (eV)
0.8
m * (me)
1.2/1.55
a (eV )
EOr ()
1.25
21
(eV)
9.0
D, (eV)
4.0
tric constant
Electron Acoustic Deformation Potential
Hole Acoustic Deformation Potential
Electron Optical Deformation Potential
Hole Optical Deformation Potential
Optical phonon energy
DA
Ref. [52]
Ref.
Ref.
11.7
Ref.
Ref.
[107]: 0.5
[60]:
+ 4.5x;
[51]: 27
[107]
10 10eV/m)
2.2(1
-
x) + 5.5x
Ref.
[51]:
2.94;
Ref.
[49]: 5.0
Ref. [109]
D,, (xl 10 10eV/m)
2.2(1
-
x) + 5.5x
Ref. [109]
D0 (x
0.0612(1- x) + 0.03704x
hw, (eV)
140
Ref. [109]
Appendix B
Derivation of the relation between
transmissivity and interface
conductance
Here we derive a relation between the transmission and reflection coefficients of an
interface and the interface conductance G, defined as:
G =
(B.1)
q
AT
where q and AT are the heat flux and temperature drop across the interface, respectively. Assuming that the incoming distributions to the interface (or forward-going
phonons at the left of the interface, and backward-going phonons at the right of the
interface) have temperatures Tei and Te2, we can use the Landauer formalism to find
the heat flux:
q=
(W)CQ,
f1 T12
Ti()4
4
ivwido
v
-w(Tei- Te2 )
(B.2)
However, it is inconsistent to define the interface conductance based on the emitted
phonon temperature. We instead need the equivalent equilibrium temperature of all
the phonons, both forward- and backward-going, at each side of the interface. Once
the equivalent equilibrium temperature is obtained, a consistent interface conductance
141
can be defined.
Here we derive the necessary relations while accounting for the
frequency dependence of the transmission coefficients.
Several approximations are necessary for this derivation. First, the known, incoming fluxes to the interface must have a thermal distribution characterized by
temperatures Tei and Te2. These will be the emitted phonon temperatures. Second,
we must linearize the equilibrium distribution; otherwise, there is no way to extract
the temperature drop from the Bose-Einstein distribution. Finally, we assume elastic,
diffuse scattering and neglect any type of mode conversion: phonons do not change
frequency or polarization as they cross the interface, but are scattered equally in all
directions.
The equilibrium distribution
f3
is given by
(w)fBE(T)
0
f Lu.(T)
4r(B.3)
= hwD
47r
where fBE(T) = (exp(hw/kBT) - 1)-1 is the Bose-Einstein distribution function, and
T is the temperature of the distribution. The linearized relation is:
fw (T)(w)
47
fBE(T)
~hwD
47
(w)
fBE
OT TO4
(T - T)
= fw (TO) + -EAT
(B.4)
We will define
foo,
= f0(To) and drop the A from AT. By the first approximation,
the incoming distribution to the interface are:
fWi-
=
=
fow i + CwiTel
47
(B.5)
+ Cw 2 Te2
(B.6)
0
We can express the outgoing distributions
f,2 and fJ-1 as the transmitted and reflected
components of these distributions. We can write the energy balance as:
j
f-1
f,+2vo 21pdyu
= T 12 (w)
fivojpdp
= T 21 (w)
j
fjiviupdy
-
R 2 1 (w)
J
fj 2 v2
c2pdy
(B.7)
f-1
f§wV2td
2 v 2 dy -
R 12 (w)
j
fjivoipdy
(B.8)
142
Because the incoming distributions are independent of angle and we assume diffuse
scattering, none of the distributions are angle-dependent. The angle integration will
just give a factor of 1/2 which will cancel. The minus sign in front of the reflection coefficients accounts for the fact that the reflected heat flux changes direction.
Inserting the thermal distributions and carrying out the angle integrations gives:
f,2vw2
f_
VW i
=
T12 (W) fo"'i + CiTei
T21 (W)
v.i + R 21 (P)
Nw2 +
Cw Te 2
2
±fow2
+
41r
)v
2
R12(W)
R2 w
C2Te2 v
(f0o1
2
(B.9)
)TeivWi (B.10)
+ C
We now have expressions for both the forward- and backward-going fluxes at each
side of the interface, which are highly non-equilibrium. The next step is to find the
equivalent equilibrium temperature on each side of the interface. To do this, we first
find the total energy density on each side of the interface. For side 1:
I (fRividTdo
+1 R2()fo1
2 (47r
w1+R12 (W)CQw1 VwiTei1
fivi+
=
~ (
+
- (T21 (W)fOw2Vw2 +T1()C2VTe
2
47r_
C
iwVTewI
(B. 11)
dw
To simplify this expression, we can use the principle of detailed balance and conservation of energy, which restricts the values of the transmission and reflection coefficients.
The principle of detailed balance requires:
T12 (W)
CiT vW i
w+
=
T21(W)
0+
Cw2 T
ow2
(B.12)
Conservation of energy requires:
R12(O)
1 -T
=
R 21 (W) =
(B.13)
12 (W)
(B.14)
1 - T 21 (W)
These equations apply at each frequency. We can use R 12 (W)
=
1 - T 12 (W) in the
second term of eq. B.11 and the principle of detailed balance in the third term. Now,
143
we also introduce the equivalent equilibrium temperature of the distribution of side
1 of the interface:
+
fifo"i
(B.15)
CoiT1
47w
Using these 3 relations, eq. B.11 becomes:
W
+-
2
(1 (
T12 (w))fo Vwi +
( 1
fo
-T1 2 (w))CiviTei
4r
+
iWTei
+ CW
+ 47w
[fv"fojW
=
+ C"Ti)v dow
- (T12 (W)/fow
2 %- %
1
+T12 (W)CwlVTe2
47r
The underscored terms cancel while the overscored terms combine. This gives:
fWl
Cwiv1Ti
JO
41
d[i
d0
T
CiVw1Tei
=
Tei
-
I
2
1 T12
2 (w)CiVw1 (Tei
-Te2)
4T)
2
47
(T 1 2 (W)Civi) (Tei - Te2 )
(Civi)
dw(B.17)
(B.18)
The procedure for side 2 of the interface is exactly the same, giving an equation for
the equivalent equilibrium temperature T 2 :
T2= Te 2 + 1 (T2 1(U)C 2V2 ) (Tei - Te2 )
2
(C2v 2 )
(B.19)
To get the interface conductance G, we need to get the equivalent equilibrium
temperature difference, obtained by subtracting the above equations:
Ti -
T2
= (Tei -Te2)
11
2
(B.20)
+
(Civi)
(C2v2)
If the transmission coefficients are frequency-independent this formula reduces to
that given in previous works.
The final step is to eliminate either T 12 (w) or T21 (w) from the equation, leaving
the interface conductance as a function of a single transmission coefficient. This can
be achieved using the principle of detailed balance. In order for this procedure to be
possible, it is necessary to use the principle of detailed balance at a single temperature
144
(B.16)
dw
To, giving
(B.21)
T12(W)/O',ivwi = T21(W)/Ow2Vwi
While this relation does depend on temperature, the temperature differences considered here are so small that it is almost exact to assume the form given above.
Then,
jW2
T2 1(w)C, 2v, 2 dW =
0
JO
T12 (w)
f~o
W T 12 (w)CliveidW
iCw2dw =
v0ow
2
(B.22)
fo
since Ci = fO,,1/fo0- 2C2. The transmissivity T 12 (w) restricts the range of the integration to those frequencies common between the two materials, making the limit of
integration irrelevant. Now, we can relate G and T 12 (w):
(T12 (w)Civi)
4
AT
1_
2
(T12(W)CiVi)
(civi)
=
+
GAT
(B.23)
(T12(W)CiVi)
(C 2 v 2 )
Solving for T 12 (w):
2
(T12 (w)Civi)
=
1
1
<>1
+ 1
(B.24)
2G
<Co>2
If we let T 12 (w) be a constant for those frequencies that transmit and zero otherwise,
we arrive at the final formula:
T12 (w)
2/|(Cv)'
2/
=
<Cv>1
1 1
±
<C>2
2G
(B.25)
where (Cv)' denotes integration over only those frequencies which have non-zero
transmissivity.
The transmissivity is a constant for those frequencies in common
between the two sides and zero otherwise. However, T 21 (w) is in general frequency
dependent because of the principle of detailed balance at each frequency.
145
146
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